Question

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 diameter 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. $\frac{\rho_{B}}{\rho_{A}}$

Answer 1

Answer: (A)

As two solid spheres are equal in masses, so
$m_{A}=m_{B}$
$\Rightarrow \frac{4}{3} \pi R_{A}^{3} \rho_{A} =\frac{4}{3} \pi R_{B}^{3} \rho_{B}$
$\Rightarrow \frac{R_{A}}{R_{B}}=\left(\frac{\rho_{B}}{\rho_{A}}\right)^{1 / 3}$
The moment of inertia of sphere about diameter
$I=\frac{2}{5} m R^{2}$
$\Rightarrow \frac{I_{A}}{I_{B}}=\left(\frac{R_{A}}{R_{B}}\right)^{2} \left(\text {as } m_{A}=m_{B}\right)$
$\Rightarrow \frac{I_{A}}{I_{B}}=\left(\frac{\rho_{B}}{\rho_{A}}\right)^{2 / 3}$