5.8.1: Potential Near a Point Mass
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We shall define the potential to be zero at infinity. If we are in the vicinity of a point mass, we shall always have to do work in moving a test particle away from the mass. We shan’t reach zero potential until we are an infinite distance away. It follows that the potential at any finite distance from a point mass is negative. The potential at a point is the work required to move unit mass from infinity to the point; i.e., it is negative.
FIGURE V.23
The magnitude of the field at a distance x from a point mass M (figure V.23) is GM/x2, and the force on a mass m placed there would be GMm/x2. The work required to move m from x to x+δx is GMmδx/x2. The work required to move it from r to infinity is
GMm∫∞rdxx2=GMmr.
The work required to move unit mass from ∞ to r, which is the potential at r is
ψ=−GMr.
The mutual potential energy of two point masses a distance r apart, which is the work required to bring them to a distance r from an infinite initial separation, is
V=−GMmr.
I here summarize a number of similar-looking formulas, although there is, of course, not the slightest possibility of confusing them. Here goes:
Force between two masses:
F=GMmr2.N
Field near a point mass:
g=GMr2,N kg−1 or m s−2
which can be written in vector form as:
g=−GMr2ˆrN kg−1 or m s−2
or as:
g=−GMr3r.N kg−1 or m s−2
Mutual potential energy of two masses:
V=−GMmr.J
Potential near a point mass:
ψ=−GMr.J kg−1
I hope that’s crystal clear.