Question

The radii of two planets are respectively $R_{1}$ & $R_{2}$ and their densities are respectively $\rho _{1}$ & $\rho _{2}$ . The ratio of the acceleration due to gravity at their surface is -

• A. $g_{1}:g_{2}=\frac{\rho _{1}}{R_{1}^{2}}:\frac{\rho _{2}}{R_{2}^{2}}$
• B. $g_{1}:g_{2}=R_{1}R_{2}:\rho _{1}\rho _{2}$
• C. $g_{1}:g_{2}=R_{1}\rho _{2}:R_{2}\rho _{1}$
• D. $g_{1}:g_{2}=R_{1}\rho _{1}:R_{2}\rho _{2}$

Answer 1

Answer: (D)

The value of $g$ at surface
$\text{g}=\frac{\text{GM}_{} \rho }{\text{R}_{}^{\text{2}}}=\frac{\text{G}}{\text{R}_{}^{\text{2}}}.\frac{\text{4\pi }}{\text{3}} \, \text{R}_{}^{\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}}$
$\Rightarrow \, \, \, \frac{g_{1}}{g_{2}} = \frac{R_{1} \rho _{1}}{R_{2} \rho _{2}}$