At what height from the surface of Earth, the gravitational potential and the value of g are -5.4 ×107 J kg-1 and 6.0 ms-2 respectively? Take the radius of Earth as 6400 km.

A
$1400 km$
B
$2000 km$
C
$2600 km$
D
$1600 km$

Solution:

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

$g_{h} = \frac{G M}{{(R + h)}^{2}} = - \frac{V_{h}}{R + h} \Rightarrow R + h = - \frac{V_{h}}{g_{h}} ∴ h = - \frac{V_{h}}{g_{h}} - R = - \frac{(- 5.4 \times 10^{7})}{6} - 6.4 \times 10^{6} = 9 \times 10^{6} - 6.4 \times 10^{6} = 2600 km$

At what height from the surface of Earth, the gravitational potential and the value of $g$ are $- 5.4 \times 10^{7} J {kg}^{- 1}$ and $6.0 {ms}^{- 2}$ respectively? Take the radius of Earth as $6400 km$ .

Solution:

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