Question

The radius of gyration about an axis through the center of a hollow sphere with external radius $a$ and internal radius $b$ is (A) \( \sqrt{\frac{2}{5}\frac{(a^{3}-b^{3})}{(a^{5}-b^{5})}} \)

• A. \( \sqrt{\frac{1}{4}\frac{(a^{4}-b^{4})}{(a^{2}-b^{2})}} \)
• B. \( \sqrt{\frac{1}{2}\frac{(a^{5}-b^{5})}{(a^{3}-b^{3})}} \)
• C. \( \sqrt{\frac{2}{5}\frac{(a^{5}-b^{5})}{(a^{3}-b^{3})}} \)
• D. \( \sqrt{\frac{5}{2}\frac{(a^{4}-b^{4})}{(a^{2}-b^{2})}} \)
• E.

Hint:

Put \(b=0\) → solid sphere, \(b=a\) → thin shell case.

Answer 1

The Correct Option is D

Concept: Radius of gyration
Radius of gyration $k$ is defined as: \[ I = Mk^2 \] where $I$ is moment of inertia and $M$ is total mass.

Step 1: Moment of inertia of hollow sphere
For a spherical shell with inner radius $b$ and outer radius $a$, the moment of inertia about its diameter is: \[ I = \frac{2}{5}M \cdot \frac{(a^5 - b^5)}{(a^3 - b^3)} \]

Step 2: Substitute in definition of $k$
\[ k = \sqrt{\frac{I}{M}} \] \[ k = \sqrt{\frac{1}{M} \cdot \frac{2}{5}M \cdot \frac{(a^5 - b^5)}{(a^3 - b^3)}} \]

Step 3: Simplify
Mass $M$ cancels: \[ k = \sqrt{\frac{2}{5}\frac{(a^5 - b^5)}{(a^3 - b^3)}} \] Final Answer:
\[ \boxed{k = \sqrt{\frac{2}{5}\frac{(a^5 - b^5)}{(a^3 - b^3)}}} \] Note:
- If $b = 0$, it reduces to solid sphere: \[ k = \sqrt{\frac{2}{5}}\,a \] - Radius of gyration represents distribution of mass about axis.