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Acceleration Due to Gravity

Two planets h...

Question

Two planets have radii $r_{1}$ and $r_{2}$ and densities $d_{1}$ and $d_{2}$ respectively. The ratio of the acceleration due to gravity on them will be

A

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

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B

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

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C

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

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D

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

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Solution

The correct option is B $r_{1} d_{1} : r_{2} d_{2}$
We have:-
$g = \frac{G M}{R^{2}}$
putting values of Density and Radius
$M = (\frac{4 π}{3} \times R^{3}) \times (d)$
$M = (V o l u m e) \times (d e n s i t y)$
$= \frac{4 π}{3} \times R^{3} d$
$g = \frac{G}{R^{2}} \times \frac{4 π}{3} \times R^{3} d$
$= \frac{4 G π}{3} \times (R d)$
$⟹ g = (C o n s t a n t) \times R d$
$\Rightarrow \frac{g_{1}}{g_{2}} = \frac{r_{1} d_{1}}{r_{2} d_{2}} = r_{1} d_{1} : r_{2} d_{2}$

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