At what distance above and below the surface of the earth a body will have same weight? (Take radius of earth as R )

Question

TestHub · Accepted Answer

The acceleration due to gravity at a height $h$ above the surface of the Earth is given by

$g_{p} = \frac{g R^{2}}{(R + h)^{2}} . . . (1)$

The acceleration due to gravity at a depth $h$ below the surface of the Earth is given by

$g_{q} = g (1 - \frac{h}{R}) . . . (2)$

According to the given problem,

$g_{p} = g_{q} . . . (3)$

Equations (1), (2) and (3) imply that

$\frac{g}{{(1 + \frac{h}{R})}^{2}} = g (1 - \frac{h}{R}) \Rightarrow (1 - \frac{h^{2}}{R^{2}}) (1 + \frac{h}{R}) = 1 . . . (4)$

Take $\frac{h}{R} = x$

So, from equation (4), it follows that

$x^{3} - x + x^{2} = 0 \Rightarrow x^{2} + x - 1 = 0 \Rightarrow x = \frac{\sqrt{5} - 1}{2} \Rightarrow h = \frac{R}{2} (\sqrt{5} - 1)$

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