Question

The radii of two planets are respectively $R_1$ and $R_2$ and their densities are respectively ${\rho }_1$ and ${\rho }_2$. The ratio of the accelerations due to gravity $\left(g_1/g_2\right)$ 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 $\frac{R_1{\rho }_1}{R_2{\rho }_2}$

Given:

Planet 1 have redius $R_1$ and density ${\rho }_1$ and Planet 2 have redius $R_2$ and density ${\rho }_2$

We know that,

$g=(\frac{GM}{R^2})$

∵ $\text{Density}=(\frac{Mass}{Volume})$

i.e. $\rho =(\frac{M}{V})⟹M=\rho V$

$⟹M=\rho (\frac{4}{3})\pi R^3$

Hence,

$g=(\frac{G}{R^2})\rho (\frac{4}{3})\pi R^3=(\frac{4}{3})\pi \rho GR$

$\therefore (\frac{g_1}{g_2})=\frac{(\frac{4}{3})\pi {\rho }_{1}G{R}_1}{(\frac{4}{3})\pi {\rho }_{2}G{R}_2}=(\frac{{\rho }_1}{{\rho }_2})(\frac{R_1}{R_2})$

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

Option B