Geology Reference
In-Depth Information
potential is defined up to an additive constant.
Generally, this constant is chosen so that
V
!
0
as
r
!1
. In the case of an infinite body, we
choose the arbitrary additive constant so that
V
D
0 at a unit distance from the body (
r
D
1).
Therefore, the
Integration over
x
0
yields:
`
2
z
0
z
0
arctan
`
1
z
0
z
0
2
3
Z
z
2
d
z
0
4
arctan
5
”
D
2G¡
z
1
solution
(
5.25
) is
changed
as
D
2G¡
I
arctan
x
0
z
0
d
z
0
follows:
(5.28)
B.S/
V.x;0;
z
/
D
G
Z
S
¡
x
0
;
z
0
where `
1
and `
2
are function of
z
0
and
B
(
S
)isthe
boundary of
S
. As before, we now approximate
the perimeter of
S
with an
N
-sided polygon
having vertices (Ÿ
1
,—
1
),(Ÿ
2
,—
2
), :::,(Ÿ
n
,—
n
). In this
case, solution (
5.28
) becomes:
(
lim
a!1
"
log
a
C
p
r
2
C
a
2
a
C
p
r
2
C
a
2
#)
dx
0
d
z
0
log
a
C
p
1
C
a
2
a
C
p
1
C
a
2
—
nC1
Z
X
N
arctan
x
0
D
2G
Z
z
0
d
z
0
¡
x
0
;y
0
log
1
”
D
2G¡
I
—
NC1
D
—
1
r
dx
0
d
z
0
(5.26)
nD1
—
n
S
(5.29)
If we move the observation point to the origin
of the reference frame and assume a constant
density ¡, then the vertical component of gravity
will be given by:
The expression for
x
0
in terms of
z
0
is similar
to (
5.20
). We have:
x
0
D
'
n
z
0
C
“
n
(5.30)
@
z
D
2G¡
Z
S
z
0
dx
0
d
z
0
x
0
2
@V
where '
n
and “
n
are given by (
5.21
). Finally,
substitution in (
5.29
) provides:
”
D
(5.27)
C
z
0
2
8
<
—
n
arctan
—
n
9
=
2
—
nC1
—
n
C
—
nC1
arctan
—
n
C
1
Ÿ
nC1
Ÿ
n
C
N
X
2
4
3
5
q
Ÿ
nC1
C
”
D
2G¡
I
Ÿ
NC1
D
Ÿ
1
I
—
NC1
D
—
1
'
n
arctan
—
nC1
—
nC1
:
;
“
n
arctan
—
n
Ÿ
n
nD1
C
log
p
Ÿ
n
C
Ÿ
nC1
1
C
'
n
—
n
(5.31)
The first two terms in parentheses of summa-
tion give zero after summation. Therefore:
8
<
9
=
2
4
3
5
q
Ÿ
n
C
1
C
—
n
C
1
p
Ÿ
n
C
—
n
'
n
arctan
—
nC1
X
N
“
n
1
C
'
n
Ÿ
nC1
arctan
—
n
”
D
2G¡
log
I
Ÿ
NC1
D
Ÿ
1
I
—
NC1
D
—
1
:
;
Ÿ
n
n
D
1
(5.32)
This solution implies that the gravity of a
body in a two-dimensional problem only depends
upon the coordinates of the vertices of a poly-
gon that approximates its cross-section. Vertex
coordinates (Ÿ
n
,—
n
)in(
5.32
) can be replaced by
quantities
r
n
and ™
n
illustrated in Fig.
5.5
. In fact,