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Consider two point masses, and , located at position vectors
and , respectively. According to Section 5.3,
the acceleration of mass due to the gravitational force exerted on it by mass takes the form

(861) 
Now, the component of this acceleration is written

(862) 
where
and
.
However, as is
easily demonstrated,







(863) 
Hence,

(864) 
with analogous expressions for and . It follows that

(865) 
where

(866) 
is termed the gravitational potential. Of course,
we can only write in the form (865) because gravity
is a conservative forcesee Chapter 2.
Note that gravitational potential, , is
directly related to gravitational potential energy, . In fact, the
potential energy of mass is
.
Now, it is wellknown that gravity is a superposable force. In other
words, the gravitational force exerted on some test mass by a collection
of point masses is simply the sum of the forces exerted on the test mass
by each point mass taken in isolation. It follows that
the gravitational potential generated by a collection of point masses
at a certain location in space is the sum of the potentials generated at that
location by each point mass taken in isolation. Hence, using Equation (866), if there are
point masses, (for ), located at position vectors ,
then the gravitational potential generated at position vector is simply

(867) 
Suppose, finally, that, instead of having a collection of point masses, we have
a continuous mass distribution. In other words, let the mass at position
vector be
, where
is the local mass density, and a volume element.
Summing over all space, and taking the limit
,
Equation (867) yields

(868) 
where the integral is taken over all space.
This is the general expression for the gravitational potential, , generated by
a continuous mass distribution, .
Next: Axially Symmetric Mass Distributions
Up: Gravitational Potential Theory
Previous: Introduction
Richard Fitzpatrick
20110331