Establish a relationship between g and G.

Let us assume a body of mass 'm' kept on the surface of the Earth of mass 'M'. The gravitational force exerted on the object equal to its weight. It is given as

F = W = mg

here g is the acceleration due to gravity.

Now according to the Newton's Law of Gravitation the force between the body and Earth is given as

F' = GMm/R2

here R is the distance between the two, which in this case will be the radius of the Earth (as body is kept on its surface) and G is the Newton's Gravitational Constant.

Now the two forces F and F' are same physically and in magnitude, thus 

F = F'


mg = GMm/R2


the relation between g and G will be

g = GM / R2

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consider earth to be sphere of mass Me and radius Re.

let whole mass of the earth is to be concentrated at the centre and a body of mass m is situated at the surface of the earth or at a small height above the surface.

so, acc. to newton's law of gravitation

F=G Me*m/Re2  1.

also,F=mass* acceleration

  =mg  2.,

from eq. 1 aqnd 2.

G Me*m/Re2 =mg

  g=G Me/Re2

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