Question

Mention the effect of altitude and depth on the value of acceleration due to gravity. Give the corresponding mathematical relation.

Answer 1

$g=\frac{GM}{R^2}$

or, $GM=g{R}^2$ $-\left(1\right)$

Now for $R\to R+h$ [$h\to$ height]

$g_h=\frac{GM}{{(R+h)}^2}$

or $GM={(R+h)}^2{g}_h$ $-\left(2\right)$

From (1) and (2)

${(R+h)}^2{g}_h=g{R}^2$

or, $g_h=g(1+\frac{2h}{R})$

Similarly for $R=R-d$ [$d\to$ depth]

$g_d=\frac{GM}{{(R-d)}^2}$

or $GM={(R-d)}^{2}gd$ $-\left(3\right)$

From (1) and (3)

$g{R}^2={(R-d)}^{2}gd$

or $g_d=g(1-\frac{d}{R})$