Physics - Gravitation Question with Solution | TestHub

The formula to calculate the mass $\left(M\right)$ of any planet of radius $R$ and density $\rho$, considering it to be a sphere, is given by

$M= \frac{4}{3}\pi R^3\rho .....................\left(1\right)$

The escape velocity $\left(v_A\right)$ for planet A can be expressed as

$\begin{array}{rcl}{V}_A& \mathit{=}& \sqrt{\frac{\mathit{2}G{M}_A}{R_A}}\\ & \mathit{=}& \sqrt{\frac{\mathit{2}G{\rho }_A{\displaystyle \frac{\mathit{4}}{\mathit{3}}}\pi {R_A}^{\mathit{3}}}{R_A}}\\ & \mathit{=}& C\sqrt{{\rho }_A\mathit{.}}{R}_A............................\left(2\right)\end{array}$

where, $G$ is Gravitational constant and $C= \sqrt{\frac{8\pi G}{3}}$ is a constant.

Similarly, the escape velocity $\left(v_B\right)$ of planet B can be expressed as

$v_B= C\sqrt{{\rho }_B}{R}_B......................\left(3\right)$

Divide equation (2) by equation (3) and solve with the known parameters to obtain the ratio of the densities of the planets.

$\begin{array}{rcl}\frac{V_A}{V_B}& \mathit{=}& \frac{\mathit{C}\sqrt{{\mathit{\rho }}_{\mathit{A}}}{\mathit{R}}_{\mathit{A}}}{\mathit{C}\sqrt{{\mathit{\rho }}_{\mathit{B}}}{\mathit{R}}_{\mathit{B}}}\\ \sqrt{\frac{{\mathit{\rho }}_{\mathit{A}}}{{\mathit{\rho }}_{\mathit{B}}}}& \mathit{=}& \mathit{ }\frac{{\mathit{V}}_{\mathit{A}}{\mathit{R}}_{\mathit{B}}}{{\mathit{V}}_{\mathit{B}}{\mathit{R}}_{\mathit{A}}}\\ \frac{{\mathit{\rho }}_{\mathit{A}}}{{\mathit{\rho }}_{\mathit{B}}}& \mathit{=}& \mathit{ }{\left(\frac{{\mathit{V}}_{\mathit{A}}{\mathit{R}}_{\mathit{B}}}{{\mathit{V}}_{\mathit{B}}{\mathit{R}}_{\mathit{A}}}\right)}^{\mathit{2}}\\ & \mathit{=}& \mathit{ }{\left(\frac{\mathit{1}}{\mathit{2}}\mathit{\times }\frac{\mathit{3}}{\mathit{1}}\right)}^{\mathit{2}}\\ & \mathit{=}& \mathit{ }\frac{\mathit{9}}{\mathit{4}}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{\left(}\mathit{4}\mathit{\right)}\end{array}$

The acceleration due to gravity $\left(g_A\right)$ for planet A is given by

$\begin{array}{rcl}{g}_A& \mathit{=}& \frac{G{M}_A}{{R_A}^{\mathit{2}}}\\ & \mathit{=}& \frac{G{\displaystyle \frac{\mathit{4}}{\mathit{3}}}\pi {R_A}^{\mathit{3}}\mathit{\times }{\rho }_A}{{R_A}^{\mathit{2}}}\\ & \mathit{=}& \mathit{ }\frac{C^2}{\mathit{2}}{\rho }_A{R}_A\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{\left(}\mathit{5}\mathit{\right)}\end{array}$

Similarly, the acceleration due to gravity $\left(g_B\right)$ can be expressed as

$g_B= \frac{C^2}{2}{\rho }_B{R}_B...........................\left(6\right)$

Divide equation (5) by equation (6) and solve with the help of known parameters to calculate the required ratio of the accelerations due to gravity in both the planets.

$\begin{array}{rcl}\frac{g_A}{g_B}& \mathit{=}& \frac{{\displaystyle \frac{C^2}{2}}{\rho }_A{R}_A}{{\displaystyle \frac{C^2}{2}}{\rho }_B{R}_B}\\ & \mathit{=}& \frac{{\rho }_A}{{\rho }_B}\frac{R_A}{R_B}\\ & \mathit{=}& \mathit{ }\frac{\mathit{9}}{\mathit{4}}\mathit{\times }\frac{\mathit{1}}{\mathit{3}}\\ & \mathit{=}& \frac{\mathit{3}}{\mathit{4}}\end{array}$

$\mathit{=}\frac{\mathit{1}}{\mathit{4}}\mathit{\times }\frac{R_{\mathit{2}}}{R_{\mathit{1}}}\mathit{=}\frac{\mathit{3}}{\mathit{4}}$