Two solid spheres ( $A$ and $B$ ) are made of metals of different densities $\rho_A$ and $\rho_B$…
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
$\left(\frac{\rho_B}{\rho_A}\right)^{2 / 3}$
$\left(\frac{\rho_A}{\rho_B}\right)^{2 / 3}$
$\frac{\rho_A}{\rho_B}$
$\frac{\rho_B}{\rho_A}$
Solution
As two solid spheres are equal in masses, so
$m_A=m_B$
$\begin{aligned} \Rightarrow & \frac{4}{3} \pi R_A^3 \rho_A & =\frac{4}{3} \pi R_B^3 \rho_B \\ \Rightarrow & \frac{R_A}{R_B} & =\left(\frac{\rho_B}{\rho_A}\right)^{1 / 3}\end{aligned}$
The moment of inertia of sphere about diameter
$I=\frac{2}{5} m R^2$
$\begin{array}{ll}\Rightarrow \quad & \frac{I_A}{I_B}=\left(\frac{R_A}{R_B}\right)^2 \quad\left(\text { as } m_A=m_B\right) \\ \Rightarrow \quad & \frac{I_A}{I_B}=\left(\frac{\rho_B}{\rho_A}\right)^{2 / 3}\end{array}$