Question

A planet has same density and same acceleration due to gravity as of earth and universal gravitational constant $G$ is twice of earth. The ratio of their radii is

• A. $1:4$
• B. $1:5$
• C. $1:2$
• D. $3:2$

Answer 1

Answer: (C)

Acceleration due to gravity at earth's surface is given by
$g=\frac{GM}{R^2}$
Since, earth is assumed to be spherical in shape, its mass is
$M=\text{ volume }\times \text{ density }=\frac{4}{3}\pi R^3\rho$
Given, ${\rho }_e={\rho }_p=\rho ,G_p=2G_e$
$\therefore \frac{g_e}{g_F}=\frac{G_e\left(\frac{4}{3}\pi R_e^3\right)\rho \times R_p^2}{R_e^2\times G_P\left(\frac{4}{3}\pi R_p^3\right)\rho }$
$1=\frac{G_e{R}_e^3\times R_p^2}{R_e^2\times R_p^3\times 2G_e}\left[\because G_p=2G_e\right]$
$1=\frac{R_e}{2R_p}$
$\Rightarrow \frac{R_p}{R_e}=\frac{1}{2}$