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

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Let the radii of the thin spherical and the solid sphere are $R_{1}$ and $R_{2}$ respectively.

Then the moment of inertia of the spherical shell about their diameter

$I = \frac{2}{3} M R_{1}^{2}$ ...(i)

and the moment of inertia of the solid sphere is given by

$I = \frac{2}{5} M R_{2}^{2}$ ...(ii)

Given that the masses and moment of inertia for both the bodies are equal, then from Eqs. (i) and (ii)

$\frac{2}{3} M R_{1}^{2} = \frac{2}{5} M R_{2}^{2} \Rightarrow \frac{R_{1}^{2}}{R_{2}^{2}} = \frac{3}{5}$

$\Rightarrow$ $\frac{R_{1}}{R_{2}} = \sqrt{\frac{3}{5}} \Rightarrow R_{1} : R_{2} = \sqrt{3} : \sqrt{5}$

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