# Variation of Acceleration due to gravity due to Altitude

## Variation of Acceleration due to gravity due to Altitude

Gravity is the force of attraction exerted by earth towards its centre on a body lying on or near the surface of earth. Gravity is merely a special case of gravitation and is also called earth’s gravitational pull. The force of attraction between two bodies on the surface of earth is the force of gravitation, but the force of attraction between each body and the earth is the force of gravity.

The acceleration due to gravity on the surface of the earth is given by:

Acceleration due to gravity (g) = GM/ R².

GM = gR² … (1)

The acceleration due to gravity at height “h” from the surface of the earth is given by:

$${{g}_{h}}=\frac{GM}{{{(R+h)}^{2}}}$$.

GM = (R + h)² gh … (2)

From equations (1) and (2), we get:

∴ R²g = (R + h)² gh

∴ $$\frac{{{g}_{h}}}{g}=\frac{{{R}^{2}}}{{{(R+h)}^{2}}}$$.

∴ $$\frac{{{g}_{h}}}{g}=\frac{{{R}^{2}}}{{{(R+h)}^{2}}}=\frac{{{R}^{2}}}{{{\left( R\left( 1+\frac{h}{R} \right) \right)}^{2}}}$$.

∴ $$\frac{{{g}_{h}}}{g}=\frac{{{R}^{2}}}{\left( {{R}^{2}}{{\left( 1+\frac{h}{R} \right)}^{2}} \right)}=\frac{1}{{{\left( 1+\frac{h}{R} \right)}^{2}}}={{\left( 1+\frac{h}{R} \right)}^{-2}}=\left( 1-\frac{2h}{R} \right)$$.

∴ $${{g}_{h}}=g\left( 1-\frac{2h}{R} \right)$$.

The above expression is for the acceleration due to gravity at small height “h” from the surface of the earth. This shows acceleration due to gravity decreases as we move away from the surface of the earth.