Acceleration due to gravity and its variation with altitude, depth and rotation of earth

Acceleration due to gravity and its variation with altitude, depth and rotation of the earth

I. Expression for acceleration due to gravity:

Consider earth to be a sphere of radius 'R' and mass 'Me'. Suppose a body of mass 'm' is placed near surface of earth then it is reasonable to assume that distance between this mass and earth is 'R' itself as whole of earth's mass can be assumed to be concentrated at its centre.

By Newton's law of gravitational force on a body of mass 'm' due to earth is

---------(1)

Since this should be equal to weight of mass 'm' on earth, that is 'mg'

Equating mg with equation (1)

we have acceleration due to gravity near surface of earth given by 

-----------(2)

This expression depicts following facts about g:

1. Value of 'g' does not depend on mass of the body (m) which is under freefall on earth
2. Since 'g' depends only on Gravitational Constant 'G' and on physical dimensions of earth, when two different masses fall freely on earth from same initial point (neglecting air resistance) both of them will reach the earth in same time given by equation s = ut + 0.5*g*t^2, irrespective of their composition, shape and size.

II. Variation of 'g' with altitude:

Consider a mass 'm' under action of earth's gravity at a height 'h' from earth's surface.

The modified 'g' now becomes, g' equals

----------(3) which is related to g from (2) as

-----------(4)

, eqn (4) may be rewritten as

g' = g/(1+h/R)^2 which can be  approximated by binomial expansion When h<<r< strong=""> as

g' = g(1-2h/R)----------(5)

From eqn (5) if h = 500 km which is much less compared to R = 6400 km then by substitution

g' = 0.84g

For h = 36000 km we have to go back to main eqn. (4) which gives

g' = (1/50)g

Hence we see that with altitude value of 'g' goes on decreasing.

II. Variation of 'g' with depth:

From eqn. (2) we know that value of 'g' was derived using assumption that earth is a sphere. In that case mass Me is given by 

 which implies by eqn (2)

--------(6)

Value of 'g' at a depth 'd' from the surface of earth can be found as follows:

Since part of earth which can be excluded from annular sphere made by 'd' is the inner sphere of radius

R-d.

Value of 'g' at surface of this inner sphere is

 --------(7) where M' = mass of the inner sphere of radius R-d given by

Using eqn. for M' in (7) we have value of 'g' at depth 'd' ;

 --------(8)

Comparing (8) with original eqn(6) we get

III. Variation of g due to rotation of earth:

Because of earth's rotation a centrifugal reaction comes into play whose magnitude is dependent on latitude 'phi' of given point P.

Effective value of 'mg' is given by parallelogram law of vectors between mg and Fcf so that