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)

$\sqrt 2 :1$

Answer 2

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
$I_{ring} = MR^2$
Moment of inertia of disc about the same axis is
$I_{disc} = \frac{MR^2}{2}$
As $I = MK^2$ where k is the radius of gyration
$\therefore I_{ring} = MK^2_{ring} = MR^2$
or $k_{ring} = R$
and $I_{disc} = Mk^2_{disc} = \frac{MR^2}{2}$
or $k_{disc} = \frac{R}{\sqrt 2}$
$\therefore \frac{k_{ring}}{k_{disc}} = \frac{R}{R/ \sqrt 2} = \frac{\sqrt 2}{1}$
$k_{ring} : k_{disc} = \sqrt 2 :1$