Geology Reference
In-Depth Information
Fig. 5.6
Geometry of a
two-dimensional problem
an
N
-sided polygon, in a way similar to that
illustrated in Fig.
5.5
. As the density of a two-
dimensional
y
dimension, we can set: ¡
D
¡(
x
,
z
). The
gravitational potential of a linear body such that
illustrated in Fig.
5.6
can be written as:
source
does
not
vary
along
the
¡
x
0
;
z
0
q
.x
x
0
/
2
V.x;0;
z
/
D
G
Z
C
.
z
z
0
/
2
dx
0
dy
0
d
z
0
C
y
02
R
2
4
3
5
D
G
Z
C
Z
¡
x
0
;
z
0
dy
0
dx
0
d
z
0
q
.x
x
0
/
2
C
.
z
z
0
/
2
C
y
02
1
S
2
3
5
dx
0
d
z
0
D
G
Z
S
Z
Ca
¡
x
0
;
z
0
dy
0
p
r
2
4
lim
a
!1
C
y
0
2
a
D
G
Z
¡
x
0
;
z
0
n
lim
a!1
h
log
a
C
p
r
2
C
a
2
log
C
a
2
io
dx
0
d
z
0
a
C
p
r
2
S
¡
x
0
;
z
0
(
lim
a!1
)
dx
0
d
z
0
p
r
2
D
G
Z
log
a
C
C
a
2
a
C
p
r
2
C
a
2
(5.25)
S
q
.x
x
0
/
2
C
.
z
z
0
/
2
and
S
is the
cross-section of the volume
R
orthogonal to the
y
axis. Clearly, as
a
!1
the limit in (
5.25
)
diverges, and the potential approaches infinity.
This
where r
definition of the potential for infinitely extended
bodies. In fact, we note that if a potential
V
the gravitational field), then
V
C
c
also satisfies
this equation for any constant
c
. Therefore, the
problem
is
overcome
by
changing
the
