Question

The radii of two planets 'A' and 'B' are 'R' and '4R' and their densities are \( \rho \) and \( \frac{\rho}{3} \) respectively. The ratio of acceleration due to gravity at their surfaces (i.e. \( g_A : g_B \)) will be:

• A. 1:16
• B. 3:16
• C. 3:4
• D. 4:3

Hint:

The ratio of accelerations due to gravity can be determined by analyzing the dependence on both the radius and the density of the planets.

Answer 1

The Correct Option is C

We are given that the radii of two planets 'A' and 'B' are \( R \) and \( 4R \), and their densities are \( \rho \) and \( \frac{\rho}{3} \), respectively. The formula for the acceleration due to gravity at the surface of a planet is: \[ g = \frac{4\pi G R \rho}{3} \] Since gravity is proportional to both the radius and density, the ratio of acceleration due to gravity at their surfaces can be written as: \[ g_A : g_B = \frac{\frac{4\pi G R \rho}{3}}{\frac{4\pi G (4R) (\frac{\rho}{3})}{3}} = \frac{R \cdot \rho}{(4R) \cdot \frac{\rho}{3}} = \frac{1}{4} \cdot 3 = \frac{3}{4} \] Thus, the correct ratio is \( g_A : g_B = 3:4 \).