Geoscience Reference
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Gm
r :
V x;y;z
Þ ¼
ð
4
:
7
Þ
ð
Apparently,
0
@
1
A ¼
9
=
V
1
r
Gm
r 2
x
r
Gm
x ¼
x
0
@
1
A ¼
V
1
r
Gm
r 2
y
r
Gm
y ¼
:
ð
4
:
8
Þ
;
y
0
1
V
1
r
Gm
r 2
z
r
Gm
@
A ¼
z ¼
z
Comparing ( 4.8 ) with ( 4.2 ), it can be seen that ( 4.8 ) gives the components of the
gravitational force ! along the three coordinate axes. This indicates that the numeric
function V in ( 4.8 ) is the gravitational potential function of a point mass.
It can be shown that the partial derivative of the potential function with respect to
an arbitrary direction is the force component in the same direction. For instance, the
partial derivative of ( 4.7 ) with respect to the direction r is:
V
Gm
r 2
r ¼
,
which has the length of the universal gravitational value and is similar to the
universal gravity value.
In order to clarify further the physical meaning of V, in Fig. 4.1 , assume that the
unit point mass m 0 moves from point B 1 (distance r 1 ) to point B 2 (distance r 2 ); then
the work (energy transfer) done by the gravitational force is:
ð
B2
B1 ¼
B2
B1
Gm
r 2
Gm
r
Gm
r 2
Gm
r 1
A
¼
dr
¼
,
where dr denotes the displacement in the direction of the force. The above equation
indicates that the potential difference between the two points is the energy needed
to move the point mass from the point of lower potential to that of higher potential.
If the potential value at point B 1 is zero, then the potential of a point equals the
energy needed to move the point mass from B 1 to this point.
Particle systems consist of a large number of point masses, and the gravitational
potential is the sum of the gravitational potentials of the masses m 1 , m 2
m n in
( 4.7 ):
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