Variation of Gravity

As I have already explained that acceleration due to gravity g is a variable quantity and it varies from place to place. We here, will derive the relation that describes variation of g with altitude.

Let us consider a body of mass M lying on the surface of earth of mass M and radius R. Let g be value of acceleration due to gravity on the free surface of earth.

Then

…………….. (i)

Suppose the body is taken to height ‘h' above the surface of earth where the value of acceleration due to gravity is gh.

Then

…………(ii)

Where (R+h) is the distance between the centers of body and earth.

Dividing (ii) by (i), we get

when h < < R, then

Since, h < < R, then h/r is very small as compared to I. Expanding the right hand side of the above equation by Binomial Theory and neglecting squares and higher powers of h/r, we get

Thus, acceleration due to gravity decreases with increase in height / altitude.