Physics -- Gravitational Field

This is the problem of taking reference point for calculating the gravitational potential energy.
For PE = mgh (CE level), the reference point of calculation is at the Earth's surface.
For GPE = -GMm/r, the reference point of calculation is at the centre of the Earth.
And if we want to compare the compatibility between 2 equations, we should note that it is valid only for a height much smaller than the Earth's radius which can be manipulated as follows:
Let M = mass of earth, m = mass of object, r = radius of earth
GPE value at ground level = -GMm/r
GPE value at a height h (<<r) above ground = -GMm/(r + h)
Then, the difference between the potential energy values at 2 levels is:
-GMm/(r + h) - (-GMm/r)
= GMm[1/r - 1/(r + h)]
= GMmh/r(r + h)
≒ GMmh/r2 (∵ h << r)
= mgh (By relation g at surface = Gm/r2)
And the result follows.
On the other hand, conceptually, the equations can also be explained as follows:

For PE = mgh, its physical meaning is the difference between gravitational potential energy at 2 different heights (Still, h << radius of Earth)
For GPE = -GMm/r, its physical meaning is the work done by an external force agent on the object when being brought from infinity to a certain distance from the centre of the Earth under the condition of keeping constant velocity (That's why negative since the external force is in opposite direction to the motion of the object).


2008-01-29 16:59:59 補充：
Just simply:If h is very small compared to radius of the planet (Not necessarily the earth), then we use mghOtherwise we use GPE = -GMm/rIn fact if h is 1% of r or smaller, then we can regard h