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. $\sqrt{\frac{2}{5} \frac{\left(a^{5}-b^{5}\right)}{\left(a^{3} -b^{3}\right)}}$

Answer 1

Answer: (D)

$\sqrt{\frac{2}{5} \frac{\left(a^{5}-b^{5}\right)}{\left(a^{3} -b^{3}\right)}}$

Answer 2

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,
$K=\sqrt{\frac{I}{M}}$

$\Rightarrow K=\frac{\sqrt{\frac{2}{5} M\left(\frac{a^{5}-b^{5}}{a^{3}-b^{3}}\right)}}{M}$

$\Rightarrow K=\sqrt{\frac{2}{5}\left(\frac{a^{5}-b^{5}}{a^{3}-b^{3}}\right)}$

So, the correct option is (D): $\sqrt{\frac{2}{5}\left(\frac{a^{5}-b^{5}}{a^{3}-b^{3}}\right)}$