Question

The radii of two planets are $R_1$ and $R_2$ and their densities are ${\rho }_1$ and ${\rho }_2$ respectively. If $g_1$ and $g_2$ represent the acceleration due to gravity at their respective surfaces, then $\frac{g_1}{g_2}$ is

• A. $\frac{{\rho }_1{R}_2^2}{{\rho }_2{R}_1^2}$
• B. $\frac{{\rho }_1{R}_1^2}{{\rho }_2{R}_2^2}$
• C. $\frac{{\rho }_2{R}_1}{{\rho }_1{R}_2}$
• D. $\frac{{\rho }_1{R}_1}{{\rho }_2{R}_2}$

Answer 1

Answer: (D)

The value of $g$ at surface
$\text{g}=\frac{{\text{Gm}}_{\text{e}}}{{\text{R}}_{\text{e}}^{\text{2}}}=\frac{\text{G}}{{\text{R}}_{\text{e}}^{\text{2}}}.\frac{4\pi }{\text{3}}{\text{R}}_{\text{e}}^{\text{3}}\rho$
$g=\frac{4}{3}\pi GR\rho$
So, $\frac{g_1}{g_2}=\frac{\frac{4}{3}\pi G{R}_1{\rho }_1}{\frac{4}{3}\pi G{R}_2{\rho }_2}\Rightarrow \frac{g_1}{g_2}=\frac{R_1{\rho }_1}{R_2{\rho }_2}$