At what height from the surface of earth the gravitation potential and the value of are and respectively? Take the radius of earth as 6400 km. [2016]

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Solution

Q.
At what height from the surface of earth the gravitation potential and the value of $g$ are $- 5.4 \times 10^{7}$ $J$ ${kg}^{- 1}$ and $6.0$ $m s^{- 2}$ respectively Take the radius of earth as 6400 km. [2016]

1 1400 km

2 2000 km

3 2600 km

4 1600 km

Ans.
(3)

Gravitation potential at a height $h$ from the surface of earth, $V_{h} = - 5.4 \times 10^{7}$ $J$ ${kg}^{- 1}$

At the same point, acceleration due to gravity, $g_{h} = 6$ $m$ $s^{- 2}$

$R = 6400 km = 6.4 \times 10^{6} m$

We know, $V_{h} = - \frac{G M}{(R + h)},$

$g_{h} = \frac{G M}{{(R + h)}^{2}} = - \frac{V_{h}}{R + h}$

$\Rightarrow R + h = - \frac{V_{h}}{g_{h}}$

$∴$ $h = - \frac{V_{h}}{g_{h}} - R = - \frac{(- 5.4 \times 10^{7})}{6} - 6.4 \times 10^{6}$

$= 9 \times 10^{6} - 6.4 \times 10^{6} = 2600$ $km$

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