Geology Reference
In-Depth Information
y
( x, y, z )
Plan
x
θ
x
r
x
ρ
δ
g =
δ
g z
z
r
δ
x'
δ
y'
Section
δ
z'
m
( x', y', z' )
Fig. 6.7 Coordinates describing an infinite horizontal line mass.
or
z
Gmz
r
Fig. 6.8 The gravity anomaly of an element of a mass of
irregular shape.
D g
=
(6.6)
3
Since a sphere acts as though its mass were concentrated
at its centre, equation (6.6) also corresponds to the grav-
ity anomaly of a sphere whose centre lies at a depth z .
Equation (6.6) can be used to build up the gravity
anomaly of many simple geometric shapes by construct-
ing them from a suite of small elements which corre-
spond to point masses, and then summing (integrating)
the attractions of these elements to derive the anomaly of
the whole body.
Integration of equation (6.6) in a horizontal direction
provides the equation for a line mass (Fig. 6.7) extending
to infinity in this direction
D g
= 2 pr
Gt
(6.8)
where r is the density of the slab and t its thickness. Note
that this attraction is independent of both the location
of the observation point and the depth of the slab.
A similar series of integrations, this time between
fixed limits, can be used to determine the anomaly of a
right rectangular prism.
In general, the gravity anomaly of a body of any shape
can be determined by summing the attractions of all the
mass elements which make up the body. Consider a small
prismatic element of such a body of density r , located at
x ¢, y ¢, z ¢, with sides of length d x ¢, d y ¢, d z ¢ (Fig. 6.8).The
mass d m of this element is given by
2
Gmz
r
D g
=
(6.7)
2
Equation (6.7) also represents the anomaly of a horizon-
tal cylinder, whose mass acts as though it is concentrated
along its axis.
Integration in the second horizontal direction pro-
vides the gravity anomaly of an infinite horizontal sheet,
and a further integration in the vertical direction be-
tween fixed limits provides the anomaly of an infinite
horizontal slab
mx
y
¢
z
¢
d
d
d
d
Consequently, its attraction d g at a point outside the
body ( x , y , z ), a distance r from the element, is derived
from equation (6.6):
gG zz
r
¢-
(
)
=
r
xyz
¢¢ ¢
d
d
d
d
3
 
Search WWH ::




Custom Search