Question

Find the ratio of radii of gyration of a circular disc and a circular ring of same mass and radius, about an axis passing through their centre and perpendicular to their planes.

• A. $1:\sqrt{2}$
• B. $3:2$
• C. $2:1$
• D. $\sqrt{2}:1$

Answer 1

The correct option is A $1:\sqrt{2}$

Let $m$ and $r$ be the mass and radius of the ring and disc respectively.
Moment of inertia of ring about an axis passing through its centre and perpendicular to its plane:
$I_{\text{ring}}=m{r}^2$

Moment of inertia of disc about an axis passing through its centre and perpendicular to its plane:
$I_{\text{disc}}=\frac{m{r}^2}{2}$
Since, $I=m{K}^2$ where, $K$ is radius of gyration about the given axis.
$K_{\text{ring}}=\sqrt{\frac{I_{\text{ring}}}{m}}=r$
$K_{\text{disc}}=\sqrt{\frac{I_{\text{disc}}}{m}}=\frac{r}{\sqrt{2}}$
$\frac{K_{\text{disc}}}{K_{\text{ring}}}=\frac{1}{\sqrt{2}}$
$\therefore K_{\text{disc}}:K_{\text{ring}}=1:\sqrt{2}$