Question

Derive the relationship between `"g"` and `"G"`?

Answer 1

`GM=gr^2`
r is the distance between object and the center of earth `r=h+R`
h is height of object from the surface and R is the radius of earth.
and
`g=4/3 piGRrho`

Explanation:

Suppose Earth is a sphere of radius `r`. It has mass `M`.
Gravitational Force on the object of mass `m` which is situated at a distance `r` from the center of Earth is
`F=(GMm)/r^2` . . .[A]

If the object is free falling from the height h from the surface of earth (or at a distance `r` from the center of earth) it experience the acceleration `g`

According to Newton's second law
Force on the object due to acceleration `g` is
`F=mg` . . .[B]

Comparing Equation [A] and [B]
`cancelmg=(GMcancelm)/r^2`
`gr^2=GM`
Where `r=h+R`
`R="Radius of Earth"`
`h="height of object from the surface of earth"`

If `h`<<`R`
then we can write `r=R+happroxR`
or
`gR^2=GM` . . .[C]

(When the object is near the surface of earth so we can neglect the height of object comparing with Radius of Earth)

If average density of earth is `rho`
then mass of earth `M="Volume"xx"density"`
`M=4/3piR^3rho`

In equation [c]
`gR^2=G4/3piR^3rho`
`g=(G4/3piR^3rho)/R^2`
`g=4/3piGRrho`