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(C) Let the radii of the thin spherical shell and the solid sphere be $R_{1}$ and $R_{2}$ respectively.The moment of inertia of the thin spherical she

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$\sqrt{3}: \sqrt{5}$

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(C) Let the radii of the thin spherical shell and the solid sphere be $R_{1}$ and $R_{2}$ respectively.The moment of inertia of the thin spherical shell about its diameter is given by $I_{1} = \frac{2}{3} M R_{1}^{2}$.The moment of inertia of the solid sphere about its diameter is given by $I_{2} = \frac{2}{5} M R_{2}^{2}$.Given that the masses $M$ and the moments of inertia $I$ are equal for both bodies,we have $I_{1} = I_{2}$.Therefore,$\frac{2}{3} M R_{1}^{2} = \frac{2}{5} M R_{2}^{2}$.Canceling $M$ and the factor $2$ from both sides,we get $\frac{R_{1}^{2}}{3} = \frac{R_{2}^{2}}{5}$.Rearranging the terms,we get $\frac{R_{1}^{2}}{R_{2}^{2}} = \frac{3}{5}$.Taking the square root on both sides,we get $\frac{R_{1}}{R_{2}} = \sqrt{\frac{3}{5}} = \frac{\sqrt{3}}{\sqrt{5}}$.Thus,the ratio of their radii is $\sqrt{3}: \sqrt{5}$.

Two spheres of equal masses,one of which is a thin spherical shell and the other a solid sphere,have the same moment of inertia about their respective diameters. The ratio of their radii will be

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