Question

Derive the relation between g and G.

Answer 1

The gravitational force applied by the earth on an object of mass m kept on its surface is given by,
$F=G\frac{Mm}{r^2}$

The weight of the object of mass m on the surface of the earth is given by,

$F=mg$

where $g$ is the acceleration due to gravity.
The weight of the object is equal to the gravitational force applied by the earth on it.

Therefore,

$mg=G\frac{Mm}{r^2}$
Divide both sides by m
$g=G\frac{M}{r^2}$

In above g and G relation :
g = acceleration due to gravity of planet
G = universal gravitational constant
M = mass of planet
r = radius of planet