Prove that at points near the surface of the Earth, the gravitational potential energy of the object is U = mgh.

Explanation

When an object of mass m is raised to a height h, the potential energy stored in the object is mgh. This can be derived using the general expression for gravitational potential energy. Consider the Earth and mass system, with r, the distance between the mass m and the Earth.s centre. Then the gravitational potential energy.
`U = - (GM_e m )/(r) ` ....(1)

Here `r = R_e + h` , where `R_e` is the radius of the Earth, h is the height above the Earth.s surface
`U = - G (M_e m)/((R_e + h) ) ` ...(2)
If h < < `R_e` , equation (2) can be modified as
`U = - G (M_e m)/(R_e (1 + h/(R_e) ) )`
`U = -G (M_e m)/(R_e) (1 + (h)/(R_e) )^(-1)`
By using Binomial expansion and neglecting the higher order terms, we get
`(1 + x)^n = 1 + nx + (n(n-1))/(2!) x^2 + ..... + oo`
here , ` x = (h)/(R_e)` and n = - 1
`(1 + (h)/(R_e))^(-1) = (1 - (h)/(R_e) )`
Replace this value and we get,
`U = - (GM_e m)/(R_e) (1 -(h)/(R_e) )` ....(4)
We know that, for a mass m on the Earth.s surface,
` G (M_e m)/(R_e) = mg R_e` ....(5)
Substituting equation (4) in (5) we get,
` U = - mgR_e + mgh_1` ....(6)
It is clear that the first term in the above expression is independent of the height h. For example, if the object is taken from height `h_1` to `h_2` , then the potential energy at `h_1` is
`U(h_1) = - mgR_e + mgh_1` .....(7)
and the potential energy at `h_2` is
`U(h_2) = - mgR_e + mgh_2` ....(8)
The potential energy difference between `h_1` and `h_2` is
`U(h_2) - U(h_1) = mg(h_1 - h_2)` ....(9)
The term `mgR_e` in equation (7) and (8) plays no role in the result. Hence in the equation (6) the first term can be omitted or taken to zero. Thus it can be stated that the gravitational potential energy stored in the particle of mass m at a height h from the surface of the Earth is U = mgh. On the surface of the Earth, U = 0, since h is zero.