Question

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. $2:1$
• D. $\sqrt{2}:1$

Answer 1

Answer: (D)

Let $M$ and $R$ be mass and radius of the ring and the disc respectively.
Then, Moment of inertia of ring about an axis passing
through its centre and perpendicular to its plane is
$I_{\text{ring }}=M{R}^2$
Moment of inertia of disc about the same axis is
$I_{disc}=\frac{M{R}^2}{2}$
As $I=M{K}^2$ where $k$ is the radius of gyration
$\therefore I_{\text{ring }}=M{K}_{\text{ring }}^2=M{R}^2$
or $k_{\text{ring }}=R$
and $I_{\text{disc }}=M{k}_{\text{disc }}^2=\frac{M{R}^2}{2}$
or $k_{\text{disc}}=\frac{R}{\sqrt{2}}$
$\therefore \frac{k_{\text{ring }}}{k_{\text{disc }}}=\frac{R}{R/\sqrt{2}}=\frac{\sqrt{2}}{1}$
$k_{\text{ring }}:k_{\text{disc}}=\sqrt{2}:1$