Question

(a)What is the the value of $g_h$ , acceleration due to gravity at height $h$, from earth surface, How can we approximate it for small values of $h$.

Answer 1

Consider a particle of mass (m) at a height (h) from the surface of the earth. Now, the force acting on the mass due to gravity is;

$F=GMm/(R+h{)}^2$

Where M is the mass of earth and R is the radius of the earth.

The acceleration due to gravity at a certain height is ‘h’ then,

$mgh=GMm/(R+h{)}^2$

$\Rightarrow gh=GM/\left[R^2\right(1+h/R{)}^2]$

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

$g=GM/R^2$

on furthur solving we get

$g_h=g(1+h/R{)}^{-2}$

when h << R, we can apply binomial approximation:

the value of g at height ‘h’ is given by $g_h=g/(1–2h/R)$