Ratio of radius of gyration of a circular dise to that of circular ring each of same mass and radius around…
Ratio of radius of gyration of a circular dise to that of circular ring each of same mass and radius around their respective axes is
$\sqrt{2}: 1$
$\sqrt{2}: \sqrt{3}$
$\sqrt{3}: \sqrt{2}$
$1: \sqrt{2}$
Solution
Let $I_d$ and $I_r$ be the respective M.I. of the disc and the ring.
If $\mathrm{K}_{\mathrm{d}}$ and $\mathrm{K}_{\mathrm{r}}$ are the respective radii of gyration,
$\begin{array}{ll}
& \mathrm{I}_{\mathrm{d}}=\frac{1}{2} \mathrm{MR}^2=\mathrm{MK}_{\mathrm{d}}^2 \\
\therefore \quad & \mathrm{~K}_{\mathrm{d}}=\frac{\mathrm{R}}{\sqrt{2}} \\
& \mathrm{I}_{\mathrm{r}}=\mathrm{MR}^2=\mathrm{MK}_{\mathrm{r}}^2 \\
\therefore \quad & \mathrm{~K}_{\mathrm{r}}=\mathrm{R} \\
\therefore \quad & \frac{\mathrm{~K}_{\mathrm{d}}}{\mathrm{~K}_{\mathrm{r}}}=\frac{\mathrm{R}}{\sqrt{2}}=\frac{1}{\sqrt{2}}
\end{array}$