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Q.
The ratio of radii of gyration of a circular ring and a circular disc, of the same mass and radius, about an axis passing through their centres and perpendicular to their planes are :
NEETNEET 2013System of Particles and Rotational Motion
Solution:
Let $M$ and $R$ be mass and radius of the ring and the disc respectively.
Then, Moment of inertia of ring about an axis passing
through its centre and perpendicular to its plane is
$I_{\text {ring }}=M R^{2}$
Moment of inertia of disc about the same axis is
$I_{d i s c}=\frac{M R^{2}}{2}$
As $I=M K^{2}$ where $k$ is the radius of gyration
$\therefore I_{\text {ring }}=M K_{\text {ring }}^{2}=M R^{2}$
or $k_{\text {ring }}=R$
and $I_{\text {disc }}=M k_{\text {disc }}^{2}=\frac{M R^{2}}{2}$
or $k_{\text{disc}}=\frac{R}{\sqrt{2}}$
$\therefore \frac{k_{\text {ring }}}{k_{\text {disc }}}=\frac{R}{R / \sqrt{2}}=\frac{\sqrt{2}}{1}$
$k_{\text {ring }}: k_{\text{disc}}=\sqrt{2}: 1$