Two solid spheres (A and B) are made of metals having densities $\rho_A$ and $\rho_B$ respectively. If there…
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
$\left(\frac{\rho_B}{\rho_A}\right)^{2 / 3}$
$\left(\frac{\rho_A}{\rho_B}\right)^{2 / 3}$.
$\frac{\rho_A}{\rho_B}$
$\frac{\rho_B}{\rho_A}$
Solution
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) $]$