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Physics LibreTexts

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 5.23.png
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

GMmrdxx2=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 kg1 or m s2

which can be written in vector form as:

g=GMr2ˆrN kg1 or m s2

or as:

g=GMr3r.N kg1 or m s2

Mutual potential energy of two masses:

V=GMmr.J

Potential near a point mass:

ψ=GMr.J kg1

I hope that’s crystal clear.


This page titled 5.8.1: Potential Near a Point Mass is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.

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