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

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Solution

Q.
At what distance above and below the surface of the earth a body will have same weight (take radius of earth as R). [2024]

1 $\sqrt{5} R - R$

2 $\frac{\sqrt{3} R - R}{2}$

3 $\frac{R}{2}$

4 $\frac{\sqrt{5} R - R}{2}$

Ans.
(4)

Below the surface of the earth, $W_{d e p t h} = \frac{G M m (R - h)}{R^{3}}$

Above the surface of the earth, $W_{a b o v e} = \frac{G M m}{{(R + h)}^{2}}$

If the body will have same weight, $g_{p} = g_{q}$

$\frac{G M m (R - h)}{R^{3}} = \frac{G M m}{{(R + h)}^{2}}$

$R^{3} = {(R + h)}^{2} (R - h) \Rightarrow h^{2} + R h - R^{2} = 0$

$\Rightarrow h = \frac{R}{2} (\sqrt{5} - 1)$

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