The magnitude of the gravitational force is given by:

Fg

= -

G m M r2

We use the connection between force and potential energy to determine the gravitational potential energy:

F

=

-

dU dr

→   U = - ∫ F dr

This gives:     U

= -

G m M r

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 = U_f - U_i = -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.