F = GMm/x^2 (Newton's law of universal gravitation)
(F = weight force, G = gravitational constant, M = earth mass, m = mass of object, x = displacement from centre of earth)
You've also been taught that:
W = Fxcos@
(W = work done, F = force, x = displacement, @ = angle between direction of force and direction of displacement)
obviously, @ = 0, because the direction of the weight force (for a perfectly spherical earth) is always towards the centre of the earth. So:
W = Fx
however, this is only valid when F is constant. In a gravitational field, F is not constant since it varies with position x. The more general definition of W is:
W = integral F dx
(sorry I can't draw up a proper integral sign)
W = integral GMm/x^2 dx
W = GMm/x/(-1) + C
W = -GMm/x + C
The only troublesome thing in there is the constant C. But look at the quantity W (work done), what does it mean? From where have we moved the object to point x, in order to do work on it?
We define the object to have zero potential energy (U), whenever x = infinite, since there is no gravitational field at this hypothetical position at all. You also know that W is simply the change in U, so it has the same dimensions:
0 = -GMm/(infinity) + C (anything divided by infinite is zero)
C = 0
hence:
U = -GMm/x
(some people like to use r instead of x, or E instead of U etc)
The reason for the minus sign is to show that there is energy stored in the object, rather than energy being applied to it.
From this result, you can determine the work done (W) to move any object from x=x0 to x=x1 using the following formula:
W = Ufinal - Uinitial = -GMm/x1 + GMm/x0
compare this to the approximation of:
W = mgh = mg(x1 - x0)
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