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

Gravitation PYQ — JEE Main 2023 · 11 Apr · Shift 1 · Q53 with Solution

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Correct answer: D

For a planet of mass $M$ and radius $R$ ,

g = \frac{GM}{R^2}

Also, mass in terms of density is:

M = \rho \cdot \frac{4}{3}\pi R^3

Substitute into the expression for $g$ :

g = \frac{G\left(\rho \cdot \frac{4}{3}\pi R^3\right)}{R^2} = \frac{4}{3}\pi G \rho R

So,

g \propto \rho R

Given:

Thus,

g_A \propto \rho R

Given:

Thus,

g_B \propto \left(\frac{\rho}{3}\right)(4R)=\frac{4\rho R}{3}

g_A : g_B = \rho R : \frac{4\rho R}{3}

Cancel $\rho R$ :

g_A : g_B = 1 : \frac{4}{3}

Multiply both terms by $3$ :

g_A : g_B = 3 : 4

Therefore, the correct answer is D.