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

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 isIring =MR2Moment of inertia of disc about the same axis isIdisc=MR22 As I=Mk2 where k is the radius of gyration ∴  Iring =Mkring 2=MR2  or   kring =R  and   Idisc =Mkdisc 2=MR22  or   kdisc =R2 ∴  kring kdisc =RR/2=21 kring :kdisc =2:1