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 $(\mathrm{g}_{\mathrm{A}}: \mathrm{g}_{\mathrm{B}})$ will be:

JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Gravitation

The radii of two planets 'A' and 'B' are 'R' and '4R' and their densities are

ρ

and

ρ /3

respectively. The ratio of acceleration due to gravity at their surfaces

(g_{A} : g_{B})

will be:

(A) $1 : 16$
(B) $3 : 16$
(C) $3 : 4$ ✓ Correct
(D) $4 : 3$

Answer: (C)

Solution

For a uniform sphere

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

with

M = ρ \cdot \frac{4}{3} π R^{3}

, so

g = \frac{4}{3} π GρR

and

g \propto ρR

. Then

\frac{g _{A}}{g _{B}} = \frac{ρ R}{( ρ /3 ) ( 4 R )} = \frac{ρR}{\frac{4}{3} ρR} = \frac{3}{4}

, that is

g_{A} : g_{B} = 3 : 4

JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Single correct option

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