The mass of a planet is that of the Earth and its diameter is half that of the Earth. The acceleration due to gravity on that planet is [2024]

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Solution

Q.
The mass of a planet is ${\frac{1}{10}}^{th}$ that of the Earth and its diameter is half that of the Earth. The acceleration due to gravity on that planet is [2024]

1 19.6 ${ms}^{- 2}$

2 9.8 ${ms}^{- 2}$

3 4.9 ${ms}^{- 2}$

4 3.92 ${ms}^{- 2}$

Ans.
(4)

Given, $M_{P} = \frac{M_{E}}{10}$ and $D_{P} = \frac{D_{E}}{2}$ or $R_{P} = \frac{R_{E}}{2}$

As,   $g_{E} = \frac{G M_{E}}{R_{E}^{2}}$ ...(i)

   $g_{P} = \frac{G M_{P}}{R_{P}^{2}}$ ...(ii)

From equation (i) and (ii)

$\frac{g_{P}}{g_{E}} = \frac{M_{P}}{R_{P}^{2}} \times \frac{R_{E}^{2}}{M_{E}} = (\frac{M_{E}}{10}) \times (\frac{4}{R_{E}^{2}}) \times \frac{R_{E}^{2}}{M_{E}}$

$∴$    $g_{P} = \frac{2}{5} g_{E} \Rightarrow g_{P} = \frac{2}{5} \times 9.8 = 3.92$ ${ms}^{- 2}$

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