The ratio of the radii of gyration of a circular disc to that of a circular ring, each of same mass and…

The ratio of the radii of gyration of a circular disc to that of a circular ring, each of same mass and radius, around their respective axes is
  1. $\sqrt{3}: \sqrt{2}$
  2. $1: \sqrt{2}$
  3. $\sqrt{2}: 1$
  4. $\sqrt{2}: \sqrt{3}$

Solution

Key Idea : The square root of the ratio of the moment of inertia of a rigid body and its mass is called radius of gyration.
As in key idea, radius of gyration is given by
$K=\sqrt{\frac{I}{M}}$
For given problem
$\frac{K_{\text {disc }}}{K_{\text {ring }}}=\sqrt{\frac{I_{\text {disc }}}{I_{\text {ring }}}}$
But $I_{\text {disc }}$ (about its axis) $=\frac{1}{2} M R^2$ and $I_{\text {ring }}$ (about its axis) $=M R^2$ where $R$ is the radius of both bodies. Therefore, Eq. (i) becomes
$\frac{K_{\mathrm{disc}}}{K_{\text {ring }}}=\sqrt{\frac{\frac{1}{2} M R^2}{M R^2}}=1: \sqrt{2}$ ~

Asked in: NEET 2008 (Screening)

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