Physics - Gravitation Question with Solution | TestHub

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

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

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.

$\begin{array}{rcl}{g}^{\text{'}}& =& \frac{GM}{R^2{\left(1+{\displaystyle \frac{h}{R}}\right)}^2}\\ & =& \frac{GM}{R^2}{\left(1+\frac{h}{R}\right)}^{-2}\\ & \approx & \frac{GM}{R^2}\left(1-\frac{2h}{R}\right) ...\left(2\right)\end{array}$

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

$g=\frac{GM}{R^2} ...\left(3\right)$

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

$\begin{array}{rcl}\frac{g\text{'}}{g}& =& \frac{\frac{GM}{R^2}\left(1-\frac{2h}{R}\right)}{\frac{GM}{R^2}}\\ & =& \left(1-\frac{{\displaystyle 2h}}{{\displaystyle R}}\right)\\ & \Rightarrow & g\text{'}=g\left(1-\frac{{\displaystyle 2h}}{{\displaystyle R}}\right)\end{array}$