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  4. Two planets A and B have the same average density. Their radii RA and RB are such that RA: RB = 3: 1. If gA and gB are the acceleration due to gravity at the surfaces of the planets, the gA: gB equals

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Q. Two planets A and B have the same average density. Their radii $R_A$ and $R_B$ are such that $R_A : R_B = 3 : 1$. If $g_A $ and $g_B$ are the acceleration due to gravity at the surfaces of the planets, the $g_A : g_{B}$ equals

KEAMKEAM 2018Gravitation

Solution:

Given, $\frac{R_{A}}{R_{B}}=\frac{3}{1}$
and $\rho_{A}=\rho_{B}$
$\because$ Average density, $\rho=\frac{3 g}{4 \pi R G}$
$\therefore $ From Eq. (ii),
$\Rightarrow \quad \frac{3 g_{A}}{4 \pi R_{A} G}=\frac{3 g_{B}}{4 \pi R_{B} G}$
$\Rightarrow \frac{g_{A}}{g_{B}}=\frac{R_{A}}{R_{B}}$
From Eq. (i), $\frac{g_{A}}{g_{B}}=\frac{3}{1}$