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

The Correct Option is 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_{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 $