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}}$