Question

Establish a relationship between g and G.

Answer 1

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

Answer 2

mg = GmM/r^2 = Weight (that is the gravitational force between particle and earth.)
Where, g is the Acceleration due to gravity
G = universal gravitational constant
M = mass of earth
m = mass of particle
r = Distance between the particle centre of mass and the centre of earth.

So,
g = GM/r^2

Hope this helps you !