The ratio of radii of gyration of a circular | vedclass

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(C) The radius of gyration $K$ is defined by the relation $I = MK^2$,where $I$ is the moment of inertia and $M$ is the mass.For a circular ring of mas

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$\sqrt{2}: 1$

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(C) The radius of gyration $K$ is defined by the relation $I = MK^2$,where $I$ is the moment of inertia and $M$ is the mass.For a circular ring of mass $M$ and radius $R$,the moment of inertia about an axis passing through its centre and perpendicular to its plane is $I_{r} = MR^2$.Thus,$MR^2 = MK_{r}^2$,which gives $K_{r} = R$.For a circular disc of mass $M$ and radius $R$,the moment of inertia about an axis passing through its centre and perpendicular to its plane is $I_{d} = \frac{1}{2}MR^2$.Thus,$\frac{1}{2}MR^2 = MK_{d}^2$,which gives $K_{d} = \frac{R}{\sqrt{2}}$.The ratio of the radii of gyration is $\frac{K_{r}}{K_{d}} = \frac{R}{R/\sqrt{2}} = \sqrt{2} : 1$.

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 is

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