QJEEAdv15Ph8ee9: JEE Advanced 2015 Physics - Class 11 JEEAdv15Ph8ee9 Solution: The densities of two solid spheres A and B of the same radii R…

Consider a spherical shell of radius r and small thickness dr.

The volume of the shell is $dV = 4\pi {r^2}dr$ and its mass is

$dm = \rho dV = 4\pi \rho {r^2}dr$ .

The moment of inertia of the spherical shell of mass dm and radius r about an axis passing through its centre O is given by $dI = {2 \over 3}dm\,{r^2}$ . Substitute the expressions for dm and $\rho$ and then integrate to get the moment of inertia of the two spheres.

${I_A} = \int_0^R {{2 \over 3}(4\pi {\rho _A}{r^4})dr = {{8\pi k} \over {3R}}\int_0^R {{r^5}dr = {{8\pi k{R^5}} \over {18}}} }$ ,

${I_B} = \int_0^R {{2 \over 3}(4\pi {\rho _B}{r^4})dr = {{8\pi k} \over {3{R^5}}}\int_0^R {{r^9}dr = {{8\pi k{R^5}} \over {30}}} }$ .

Divide to get ${I_B}/{I_A} = 6/10$ .