# 5.8.1: Potential Near a Point Mass

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- 8145

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.

\(\text{FIGURE V.23}\)

The magnitude of the field at a distance \(x\) from a point mass \(M\) (figure \(\text{V.23}\)) is \(GM/x^2\), and the force on a mass m placed there would be \(GMm/x^2\). The work required to move \(m\) from \(x\) to \(x + δx\) is \(GMmδx/x^2\). The work required to move it from \(r\) to infinity is

\[GMm∫_r^∞ \frac{dx}{x^2} = \frac{GMm}{r}.\]

The work required to move *unit* mass *from* \(∞\) to \(r\), which is the potential at \(r\) is

\[ψ = -\frac{GM}{r}. \label{5.8.1} \tag{5.8.1}\]

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

\[V = -\frac{GMm}{r}. \label{5.8.2} \tag{5.8.2}\]

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:

\[F = \frac{GMm}{r^2}. \quad \text{N} \label{5.8.3} \tag{5.8.3}\]

Field near a point mass:

\[g = \frac{GM}{r^2}, \quad \text{N kg}^{-1} \ or \text{ m s}^{-2} \label{5.8.4} \tag{5.8.4}\]

which can be written in vector form as:

\[\textbf{g} = - \frac{GM}{r^2} \hat{\textbf{r}} \quad \text{N kg}^{-1} \ or \text{ m s}^{-2} \label{5.8.5} \tag{5.8.5}\]

or as:

\[\textbf{g} = -\frac{GM}{r^3} \textbf{r}. \quad \text{N kg}^{-1} \ or \text{ m s}^{-2} \label{5.8.6} \tag{5.8.6}\]

Mutual potential energy of two masses:

\[V = -\frac{GMm}{r}. \quad \text{J} \label{5.8.7} \tag{5.8.7}\]

Potential near a point mass:

\[ψ = -\frac{GM}{r}. \quad \text{J kg}^{-1} \label{5.8.8} \tag{5.8.8}\]

I hope that’s crystal clear.