The ratio of the radii of gyration of a circular disc to that of a...

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 $\frac{K_{\mathrm{disc}}}{K_{\text{ring }}}=\sqrt{\frac{\frac{1}{2} M R^2}{M R^2}}=1: \sqrt{2}$