Two planets have radii r1 and r2 and densities d1 and d2 respectively. Then the ratio of accelerations due to gravity on them is

Question

Two planets have radii r1 and r2 and densities d1 and d2 respectively. Then the ratio of accelerations due to gravity on them is (A) r1d1: r2d2 (B) r1d2: r2d1 (C) r12d1: r22d-2 (D) r1d21: r2d22

Tardigrade Admin · Accepted Answer

Acceleration due to gravity g=(GM/r2) or g=(G/r2)×(4/3)π r3d=(4/3)π rdG = (g1/g2)=(d1r1/d2r2)

Q. Two planets have radii $r_{1}$ and $r_{2}$ and densities $d_{1}$ and $d_{2}$ respectively. Then the ratio of accelerations due to gravity on them is

$r_{1} d_{1} : r_{2} d_{2}$

$r_{1} d_{2} : r_{2} d_{1}$

$r_{1}^{2} d_{1} : r_{2}^{2} d - 2$

$r_{1} d_{1}^{2} : r_{2} d_{2}^{2}$

Acceleration due to gravity
$g = \frac{GM}{r ^{2}}$
or $g = \frac{G}{r ^{2}} \times \frac{4}{3} π r^{3} d = \frac{4}{3} π r d G$
= $\frac{g _{1}}{g _{2}} = \frac{d _{1} r _{1}}{d _{2} r _{2}}$

Q. Two planets have radii r1​ and r2​ and densities d1​ and d2​ respectively. Then the ratio of accelerations due to gravity on them is

r1​d1​:r2​d2​

r1​d2​:r2​d1​

r12​d1​:r22​d−2

r1​d12​:r2​d22​