Question

The radii of two planets are $R_1$ and $R_2$ respectively and their densities are ${\rho }_1$ and ${\rho }_2$ respectively. The ratio of the accelerations due to gravity $\frac{g_1}{g_2}$ at their surfaces is

• A. $\frac{R_1{\rho }_2}{R_2{\rho }_1}$
• B. $\frac{R_1{\rho }_1}{R_2{\rho }_2}$
• C. $\frac{{\rho }_1{R}_2^2}{{\rho }_2{R}_1^2}$
• D. $\frac{{R_{1}R}_2}{{\rho }_1{\rho }_2}$

Answer 1

The correct option is B $\frac{R_1{\rho }_1}{R_2{\rho }_2}$

Using the relation $g=\frac{GM}{R^2}$

Also $M=\rho \times V=\rho \times \frac{4}{3}\pi R^3$

Hence, $g=\frac{4}{3}G\rho \pi R$

$\frac{g_1}{g_2}=\frac{G{\rho }_1\pi R_1}{G{\rho }_2\pi R_2}$

$\frac{g_1}{g_2}=\frac{R_1{\rho }_1}{R_2{\rho }_2}$

Hence, B is the correct option.