Variation of ‘g’ with Altitude
Suppose earth is perfectly spherical in shape with uniform density ρ. We know that at the surface of earth
g = G Me / Re2
G = Gravitational constant
Me = Mass of earth
Re = Radius of earth
At a height ‘h’ above the surface of earth, gravitational acceleration is
g = G Me / (Re + h)2
Dividing (1) by (2)
g / g = [G Me / Re2] x [(Re + h)2 / G Me)
g / g = (Re + h)2 / Re2
g / g = [Re + h) / Re]2
g / g = [1 + h/Re]2
g / g = [1 + h/Re]-2
We expand R.H.S using Binomial Formula,
(1 + x)n = 1 + nx + n(n-1) x2 / 1.2 + n(n + 1)(n-2)x3 / 1.2.3 + …
If h / Re < 1, then we can neglect higher powers of h / Re.
g / g = 1 – 2 h / Re
g = g (1 – 2h / Re) …………………………… (3)
Equation (3) gives the value of acceleration due to gravity at a height ‘h’ above the surface of earth.
From (3), we can conclude that as the value of ‘h’ increases, the value of ‘g’ decreases.