Gravitational Potential Energy

The magnitude of the gravitational force is given by:
Fg = -
G m M

We use the connection between force and potential energy to determine the gravitational potential energy:
F = -
  →   U = - F dr
This gives:     U = -
G m M

if we define the potential energy to be zero at r = ∞.

Important: the potential energy is negative!

A negative potential energy is consistent with mgh for potential energy near the surface of the Earth. If you lift an object a height h from the ground, the potential energy change is:

ΔU = Uf - Ui = -GmM/(R+h) - ( -GmM/R ) .

Now use the approximation (from Taylor's theorem):

1/(R+h) = 1/[R(1+h/R)] ≈ (1/R)*(1 - h/R)

when h is small compared to R.

Substituting this in the expression for Δ U gives:

ΔU = -GmM/R + GmMh/R2 + GmM/R = GmMh/R2

We showed previously that g = GM/R2, so:    ΔU = mgh.