Q. Two solid spheres ($A$ and $B$) are made of metals of different densities $\rho_{A\ }$and $\rho_{B}$ respectively. If their masses are equal, the ratio of their moments of inertia $\left( I_{B}/I_{A} \right)$ about their respective diameters is
A) $\left( \frac{\rho_{B}}{\rho_{A}} \right)^{2/3}$
B) $\left( \frac{\rho_{A}}{\rho_{B}} \right)^{2/3}$
C) $\frac{\rho_{A}}{\rho_{B}}$
D) $\frac{\rho_{B}}{\rho_{A}}$
Explanation
(a) As two solid spheres are equal in masses, so
$m_{A} = m_{B}$
$\Rightarrow$ $\frac{4}{3}\pi R_{A}^{3}\rho_{A} = \frac{4}{3}\pi R_{B}^{3}\rho_{B}$
$\Rightarrow$ $\frac{R_{A}}{R_{B}} = \left( \frac{\rho_{B}}{\rho_{A}} \right)^{\frac{1}{3}}$
The moment of inertia of sphere about diameter
$I = \frac{2}{5}mR^{2}$
$\Rightarrow$ $\frac{I_{A}}{I_{B}} = \left( \frac{R_{A}}{R_{B}} \right)^{2}$ (as $m_{A} = m_{B}$)
$\Rightarrow$ $\frac{I_{A}}{I_{B}} = \left( \frac{\rho_{B}}{\rho_{A}} \right)^{\frac{2}{3}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.