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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.
g = GM / R2 …………….. (i)
Suppose the body is taken to height ‘h' above the surface of earth where the value of acceleration due to gravity is gh.
gh = GM / (R + h )2 …………(ii)
Where (R+h) is the distance between the centers of body and earth.
Dividing (ii) by (i), we get
gh / g = GM / (R + h )2 X R2 / GM = R2 / (R+ h ) 2
when h < < R, then
gh / g = R2 / R2 ( 1 + h2 / R2 ) 2
= 1/ ( 1 + h2 / R2 ) 2
= ( 1 + h2 / R2 ) -2
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
gh / g = 1-2h / R
Thus, acceleration due to gravity decreases with increase in height / altitude.