The acceleration due to gravity at height above the earth, if (radius of earth), is given by [2023]

Question

The acceleration due to gravity at height $h$ above the earth, if $h ≪ R$ (radius of earth), is given by [2023]

Smart Achievers · Accepted Answer

(1)

$For a point outside the surface of Earth, g = \frac{G M}{r^{2}}$

$r = distance from centre of Earth$

$\Rightarrow g (h) = \frac{G M}{{(R + h)}^{2}} = \frac{G M}{R^{2} {(1 + \frac{h}{R})}^{2}}$

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

$If h ≪ R, {(1 + \frac{h}{R})}^{- 2} \approx 1 - \frac{2 h}{R}$

$\Rightarrow g (h) = \frac{G M}{R^{2}} (1 - \frac{2 h}{R})$

$\Rightarrow g (h) = g_{surface} (1 - \frac{2 h}{R}), \frac{G M}{R^{2}} = g_{surface}$

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