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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 $6.0\ ms^{-2}$ respectively? (Take, the radius of earth as 6400 km.)
A
1600 km
B
1400 km
C
2000 km
D
2600 km
Explanation
$|V|/g' = R + h$.
Detailed Solution
Gravitational potential at height h: $V = -\frac{GM}{R + h}$ ...(i)
Gravitational acceleration at height h: $g' = \frac{GM}{(R + h)^2}$ ...(ii)
From (i) and (ii): $\frac{|V|}{g'} = R + h \Rightarrow h = \frac{|V|}{g'} - R$
$h = \frac{5.4\times10^{7}}{6} - 6400$ km $= 0.9\times10^{7}$ m $- 6400$ km
$h = 9000$ km $- 6400$ km $= 2600$ km
Gravitational acceleration at height h: $g' = \frac{GM}{(R + h)^2}$ ...(ii)
From (i) and (ii): $\frac{|V|}{g'} = R + h \Rightarrow h = \frac{|V|}{g'} - R$
$h = \frac{5.4\times10^{7}}{6} - 6400$ km $= 0.9\times10^{7}$ m $- 6400$ km
$h = 9000$ km $- 6400$ km $= 2600$ km