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
$g=\frac{G M}{R^{2}}$
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}=2 G_{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 2 G_{e}} \left[\because G_{p}=2 G_{e}\right] $
$ 1 =\frac{R_{e}}{2 R_{p}} $
$ \Rightarrow \frac{R_{p}}{R_{e}} =\frac{1}{2} $