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

Answer 1

Answer: (D)

As we know moment of inertia of a hollow sphere with external radius $a$ and internal radius $b$ is
$I=\frac{2}{5}M\left(\frac{a^5-b^5}{a^3-b^3}\right)$
Radius of gyration, $K=\sqrt{\frac{I}{M}}$
where, $M=$ mass of sphere.
so, $K=\frac{\sqrt{\frac{2}{5}M\left(\frac{a^5-b^5}{a^3-b^3}\right)}}{M}$
$\Rightarrow K=\sqrt{\frac{2}{5}\left(\frac{a^5-b^5}{a^3-b^3}\right)}$