Question 531: Centre of Mass and Rotational Motion

Question

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 $\left(\frac{\mathrm{I}_{\mathrm{B}}}{\mathrm{I}_{\mathrm{A}}}\right)$ about their respective diameter is

Accepted Answer

Mass $=$ Volume $	imes$ Density
Mass of spheres $A$ and $B$ is,
$\mathrm{M}_{\mathrm{A}}=\frac{4}{3} \pi \mathrm{R}_{\mathrm{A}}^3 ho_{\mathrm{A}}$ and $\mathrm{M}_{\mathrm{B}}=\frac{4}{3} \pi \mathrm{R}_{\mathrm{B}}^3 ho_{\mathrm{B}}$
If masses are equal,
$\mathrm{M}_{\mathrm{A}}=\mathrm{M}_{\mathrm{B}}$
$$\begin{array}{ll}
	herefore \quad & \frac{4}{3} \pi \mathrm{R}_{\mathrm{A}}^3 ho_{\mathrm{A}}=\frac{4}{3} \pi \mathrm{R}_{\mathrm{B}}^3 ho_B \
& \frac{\mathrm{R}_B}{\mathrm{R}_{\mathrm{A}}}=\left(\frac{ho_{\mathrm{A}}}{ho_B}ight)^{\frac{1}{3}}\quad	ext{..... (i)}
\end{array}$$
Moment of inertia of sphere about its diameter is given by, $\mathrm{I}=\frac{2}{5} \mathrm{MR}^2$
$I_A=\frac{2}{5} M_A R_A^2$ and $I_B=\frac{2}{5} M_B R_B^2$
$$\begin{aligned}
	herefore \quad & \frac{I_B}{I_A}=\frac{R_B^2}{R_A^2} \
& \frac{I_B}{I_A}=\left(\frac{ho_A}{ho_B}ight)^{\frac{2}{3}}\quad	ext{.... [From (i)]}
\end{aligned}$$

Answer

$\left(\frac{\rho_B}{\rho_A}\right)^{2 / 3}$

Answer

$\left(\frac{\rho_A}{\rho_B}\right)^{2 / 3}$

Answer

$\frac{\rho_A}{\rho_B}$

Answer

$\frac{\rho_B}{\rho_A}$

Centre of Mass and Rotational Motion

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{\mathrm{I}_{\mathrm{B}}}{\mathrm{I}_{\mathrm{A}}}\right) about their respective diameter is

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