Question

At what height from the surface of earth, the gravitation potential and the value of g are $- 5.4 \times 10^{7} {Jkg}^{- 2}$ and 6ms^-2, respectively? (Take, the radius of earth is 6400 km) [NEET 2016]

Moderate

Solution

Gravitational potential at some height h from the surface of the earth is given by
$V = - \frac{G M}{R + h}$
and acceleration due to gravity at some height h from the surface of the earth can be given as
$g = \frac{G M}{(R + h)^{2}}$
From Eqs. (i) and (ii), we get
$\frac{| V |}{g} = \frac{G M}{(R + h)} \times \frac{(R + h)^{2}}{G M} \Rightarrow \frac{| V |}{g} = R + h$
$∵ V = - 5.4 \times 10^{7} {Jkg}^{- 2}, g = 6 {ms}^{- 2}$
and R = 6400km $= 6.4 \times 10^{6} m$
Substituting these values in Eq. (iii), we get
$\frac{5.4 \times 10^{7}}{6} = 6.4 \times 10^{6} + h$
$\Rightarrow h = (9 - 6.4) \times 10^{6} = 2.6 \times 10^{6} m \Rightarrow h = 2600 km$

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