Two solid spheres $A$ and $B$ each of radius $R$ are made of materials of densities $\rho_A$ and $\rho_B$…
Two solid spheres $A$ and $B$ each of radius $R$ are made of materials of densities $\rho_A$ and $\rho_B$ respectively. Their moments of inertia about a diameter are $I_A$ and $I_B$ respectively. The value of $\frac{I_A}{I_B}$ is
$\sqrt{\frac{\rho_A}{\rho_B}}$
$\sqrt{\frac{\rho_B}{\rho_A}}$
$\frac{\rho_A}{\rho_B}$
$\frac{\rho_B}{\rho_A}$
Solution
Moment of inertia
$\begin{aligned} I & =\frac{2}{3} M R^2 \\ \therefore \quad \frac{I_A}{I_B} & =\frac{\frac{2}{5} M_A R^2}{\frac{2}{5} M_B R^2}=\frac{\frac{4}{3} \pi R^3 \rho_A}{\frac{4}{3} \pi R^3 \rho_B}=\frac{\rho_A}{\rho_B}\end{aligned}$