Two planets of radii $r_{1}$ and $r_{2}$ are made from the same material.The ratio of the acceleration of gravity $\frac{g_{1}}{g_{2}}$ at the surface of the two planets is-

\frac{r_{1}}{r_{2}}

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\frac{r_{2}}{r_{1}}

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(r_{1} / r_{2})^{2}

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(r_{2} / r_{1})^{2}

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Solution

The correct option is A $\frac{r_{1}}{r_{2}}$
We know that acceleration due to gravity can be described as $g = \frac{G m}{R^{2}}$
Mass of planet $(m_{1}) = d e n s i t y \times v o l u m e$
$= d \times \frac{4}{3} π r_{1}^{3}$
$\Rightarrow m_{2} = d \times \frac{4}{3} π r_{2}^{3}$
Now, $g_{1} = \frac{4 (G \times d \times π r_{1}^{3})}{3 r_{1}^{2}} = \frac{4}{3} G \times d \times r_{1}$
$g_{2} = \frac{4 (G \times d \times π r_{2}^{3})}{3 r_{2}^{2}} = \frac{4}{3} G \times d \times r_{2}$
Ratio of acceleration due to gravity we get,
$\Rightarrow \frac{g_{1}}{g_{2}} = \frac{\frac{4}{3} \times G \times d \times r_{1}}{\frac{4}{3} \times G \times d \times r_{2}} = \frac{r_{1}}{r_{2}}$
Hence, the answer is $\frac{r_{1}}{r_{2}} .$

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