The acceleration due to gravity at height h above the earth if h ≪...

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

${g}^{'} = \dfrac{GM}{( R + h{)}^{2}} . . . \left(1\right)$

Use the method of binomial expansion to obtain the approximate expression for the acceleration due to gravity at a particular height from the surface of the Earth.

$\begin{cases}{g}^{'} & = & \dfrac{GM}{{R}^{2}{\left(1 + \dfrac{h}{R}\right)}^{2}} \\ & = & \dfrac{GM}{{R}^{2}}{\left(1 + \dfrac{h}{R}\right)}^{- 2} \\ & \approx & \dfrac{GM}{{R}^{2}}\left(1 - \dfrac{2h}{R}\right) . . . \left(2\right)\end{cases}$

The formula to calculate the acceleration due to gravity at the surface of the Earth is given by

$g = \dfrac{GM}{{R}^{2}} . . . ( 3 )$

Divide equation (2) by equation (3) and simplify to obtain the required relation.

$\begin{cases}\dfrac{g '}{g} & = & \dfrac{\dfrac{GM}{{R}^{2}}\left(1 - \dfrac{2h}{R}\right)}{\dfrac{GM}{{R}^{2}}} \\ & = & \left(1 - \dfrac{2h}{R}\right) \\ & ⇒ & g ' = g\left(1 - \dfrac{2h}{R}\right)\end{cases}$