Derive the Relation between g and G .

Explanation

The gravitational force exerted by Earth on the mass m near the surface of the Earth is given by
`vec(F) = -(GmM_E)/(R_E^2) hat(r)`
Now equating Gravitational force to Newton's second law,
`ma = -(GmM_E)/(R_E^2) hat(r)`
Hence, acceleration is,
`vec(a) = -(GM_E)/(R_E^2) hat(r)`
The acceleration experienced by the object near the surface of the Earth due to its gravity is called acceleration due to gravity. It is denoted by the symbol g. The magnitude of acceleration due to gravity is
`|g| = (GM_E)/(R_E^2)`
It is to be noted that the acceleration experienced by any object is independent of its mass. The value of g depends only on the mass and radius of the Earth.