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Gravitation — Physics STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 SciencePhysicsGravitation3 Marks
Question
While approaching a planet circling a distant star, a space traveller determines the planet's radius to be half that of the earth. After landing on the surface, he finds the acceleration due to gravity to be twice that on the surface of the earth. Find the ratio of the mass of the planet to that of the earth.
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Answer

In case of the earth, $\frac{\text{GM}_{\text{e}}\text{m}}{\text{r}^2_{\text{e}}}=\text{mg}_{\text{e}}$
In case of the planet, $\frac{\text{GM}_{\text{p}}\text{m}}{\text{r}^2_{\text{p}}}=\text{mg}_{\text{p}}$
Dividing these two equations, we get
$\Big(\frac{\text{M}_{\text{p}}}{\text{M}_{\text{e}}}\Big)\Big(\frac{\text{r}^2_{\text{e}}}{\text{r}^2_{\text{p}}}\Big)=\frac{\text{g}_{\text{p}}}{\text{g}_{\text{e}}},$
$\text{but }\text{g}_{\text{p}}=2\text{g}_{\text{e}}\text{ and }\text{r}_{\text{p}}=\frac{\text{r}_{\text{e}}}{2}$
$\therefore\frac{\text{M}_{\text{p}}}{\text{M}_{\text{e}}}=\frac{2}{4}=\frac{1}{2}$
Thus the ratio of the mass of the planet to the mass of the earth is $\frac{1}{2}.$

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