Two planets have radius in the ratio $x : y$ and density in the ratio $m : n .$ The acceleration due to gravity on the surface of the two planets will be in the ratio of

n x : m y

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n y : m x

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m x : n y

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m y : n x

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Solution

The correct option is C $m x : n y$
Let the two planets be A and B

$\begin{matrix} Planet A & Planet B Radius & R_{A} & R_{B} Density & D_{A} & D_{B} Mass & M_{A} & M_{B} Acceleration d u e t o g r a v i t y & g_{A} & g_{B} \end{matrix}$

Acceleration due to gravity of planet A, $g_{A} = \frac{G M_{A}}{R_{A}} g_{A} = \frac{G D_{A} \times \frac{4}{3} π R_{A}^{3}}{R_{A}^{2}} = G D_{A} . \frac{4}{3} π R_{A} (∵ M a s s = D e n s i t y \times V o l u m e = D \times \frac{4}{3} π R^{3}) g_{A} = \frac{4}{3} π G_{A} R_{A} D_{A} . . . . (i)$

Similarly, acceleration due to gravity of planet $B, g_{B} = \frac{G M_{B}}{R_{B}^{2}} g_{B} = \frac{4}{3} π G D_{B} R_{B} . . . . . . (i i)$

According to the question:
$\frac{R_{A}}{R_{B}} = \frac{x}{y}, \frac{D_{A}}{D_{B}} = \frac{m}{n}$

Taking ratio of (i) and (ii):

$\frac{g_{A}}{g_{B}} = \frac{D_{A}}{D_{B}} . \frac{R_{A}}{R_{B}} \frac{g_{A}}{g_{B}} = \frac{m}{n} . \frac{x}{y} g_{A} : g_{s} = m x : n y$

Hence, the correct answer is option (c).

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Two planets have radius in the ratio x:y and density in the ratio m:n. The acceleration due to gravity on the surface of the two planets will be in the ratio of

Two planets have radius in the ratio $x : y$ and density in the ratio $m : n .$ The acceleration due to gravity on the surface of the two planets will be in the ratio of