The ratio of radii of gyration of a circular ring and a circular...

Let $M and R$ be the mass and radius of ring and disc then M.I. of ring about an axis passing through its centre and perpendicular to its plane is I_${\text{ring }}$=M R² And M.I. of disc about an axis passing through centre is \begin{aligned} & I_{\text{disc }}=\frac{M R^2}{2} \\ & \text{ i.e. } \quad \begin{aligned} I & =M K^2 \\ \text{ where } K=\text{ radius of } & \text{ gyration } \\ \therefore \quad I_{\text{ring }} & =M K_{\text{ring }}^2 \\ & =M R^2 \\ I_{\text{fring }} & =R \\ I_{\text{disc }} & =M K_{\text{disc }}^2 \\ & =\frac{M K^2}{2} \\ K_{\text{disc }} & =\frac{R}{\sqrt{2}}\end{aligned}\end{aligned}$$\begin{aligned} \frac{\mathrm{K}_{\text{ring }}}{\mathrm{K}_{\text{desc }}} & =\frac{R}{\frac{R}{\sqrt{2}}}=\frac{\sqrt{2}}{1} \\ & =\sqrt{2}: 1\end{aligned}$$$