First slide

Moment of intertia

Question

The ratio of the radii of gyration of a hollow sphere and a solid sphere of the same radii about a tangential axis is

Easy

A

$\sqrt{\frac{7}{3}}$
B

$\frac{5}{\sqrt{21}}$
C

$\sqrt{\frac{21}{5}}$
D

$\frac{25}{9}$

Solution

For hollow sphere, $K_{H} = \sqrt{\frac{I}{m}} = \sqrt{\frac{\frac{2}{3} m R^{2} + m R^{2}}{m}} = \sqrt{\frac{5}{3}} R$
For solid sphere, $K_{S} = \sqrt{\frac{I}{m}} = \sqrt{\frac{\frac{2}{5} m R^{2} + m R^{2}}{m}} = \sqrt{\frac{7}{5}} R$
$∴ \frac{K_{H}}{K_{S}} = \sqrt{\frac{25}{21}} = \frac{5}{\sqrt{21}}$

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