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Q.
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
MHT CETMHT CET 2008System of Particles and Rotational Motion
Solution:
Let the radii of the thin spherical shell and the solid sphere are $R_1$ and $R_2$ respectively
Then the moment of inertia of the spherical shell about their diameter
$I=\frac{2}{3}MR^2_1$ ......(i)
and the moment of inertia of the solid sphere is given by
$I=\frac{2}{5}MR^2_2$ ......(ii)
It is given that the masses and moment of inertia for both the bodies are equal, then from Eqs. (i) and (ii)
$\frac{2}{3}MR^2_1=\frac{2}{5}MR^2_2$
$\Rightarrow \frac{R^2_1}{R^2_2}=\frac{3}{5}$
$\Rightarrow \frac{R_1}{R_2}=\sqrt{\frac{3}{5} }$
$\Rightarrow R_1:R_2=\sqrt{3}: \sqrt{5}$