Q102 / 180 · NEET 2013 (Karnataka) · Physics
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:
- A$1: \sqrt{2}$
- B$3: 2$
- C$2: 1$
- Correct answerD$\sqrt{2}: 1$
Solution
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
$$
I_{\text {ring }}=M R^2
$$
And M.I. of disc about an axis passing through centre is
$\begin{aligned} & I_{\text {disc }}=\frac{M R^2}{2} \\ & \text { i.e. } \quad \begin{aligned} I & =M K^2 \\ \text { where } K=\text { radius of } & \text { gyration } \\ \therefore \quad I_{\text {ring }} & =M K_{\text {ring }}^2 \\ & =M R^2 \\ I_{\text {fring }} & =R \\ I_{\text {disc }} & =M K_{\text {disc }}^2 \\ & =\frac{M K^2}{2} \\ K_{\text {disc }} & =\frac{R}{\sqrt{2}}\end{aligned}\end{aligned}$
$\begin{aligned} \frac{\mathrm{K}_{\text {ring }}}{\mathrm{K}_{\text {desc }}} & =\frac{R}{\frac{R}{\sqrt{2}}}=\frac{\sqrt{2}}{1} \\ & =\sqrt{2}: 1\end{aligned}$
Hence, the correct answer is option: 'D'.