Acceleration due to gravity and its variation with altitude, depth and rotation of 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 gravitation attract ive 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