Two solid spheres (A and B) are made of metals having densities \(\rho_A\) and \(\rho_B\) respectively. If there masses are equal then ratio of their moments of inertia \(\left(\frac{I_B}{I_A}\right)\) about their respective diameter is

Mass \(=\) Volume \(\times\) Density
Mass of spheres \(A\) and \(B\) is, \(\mathrm{M}_{\mathrm{A}}=\frac{4}{3} \pi \mathrm{R}_{\mathrm{A}}^3 \rho_{\mathrm{A}}\) and \(\mathrm{M}_{\mathrm{B}}=\frac{4}{3} \pi \mathrm{R}_{\mathrm{B}}^3 \rho_{\mathrm{B}}\)
If masses are equal,
\(\mathrm{M}_{\mathrm{A}}=\mathrm{M}_{\mathrm{B}}\)
\(\therefore \quad \frac{4}{3} \pi \mathrm{R}_{\mathrm{A}}^3 \rho_{\mathrm{A}}=\frac{4}{3} \pi \mathrm{R}_{\mathrm{B}}^3 \rho_{\mathrm{B}}\)
\(\frac{R_B}{R_A}=\left(\frac{\rho_A}{\rho_B}\right)^{\frac{1}{3}}\)
Moment of inertia of sphere about its diameter is given by, \(\mathrm{I}=\frac{2}{5} \mathrm{MR}^2\)
\(\begin{array}{ll}
\quad & I_A=\frac{2}{5} M_A R_A^2 \text { and } I_B=\frac{2}{5} M_B R_B^2 \\
\therefore \quad & \frac{I_B}{I_A}=\frac{R_B^2}{R_A^2}
\end{array}\)
\(\frac{I_B}{I_A}=\left(\frac{\rho_A}{\rho_B}\right)^{\frac{2}{3}}\)
\(\ldots[\) From (i) \(]\)