Geology Reference
In-Depth Information
L
ê
3
b-g
w
b
g
Figure 4.10 The three coplanar vectors: the unit vector
e
3
in the direction of the
reference axis, the angular momentum vector
L
and the angular velocity vector
ω
of the rotation.
from equation (4.82), we have that
|
|
A
L
L
A
=Ω+
σ
0
,
cos
(
β
−
γ
)
=
e
3
·
(4.86)
neglecting squares of the small quantities
m
1
,
m
2
.Wealsohavethat
|
A
=
(
ω
L
|
+
σ
0
e
3
)
1/2
+
σ
0
e
3
)
·
(
ω
ω
0
1/2
=
·
ω
+
2σ
0
ω
·
e
3
+
σ
.
(4.87)
From equation (4.84),
2
1
β
,
2
2
sec
2
tan
2
ω
=Ω
β
=
ω
·
ω
=Ω
+
|
|
(4.88)
using the trigonometric identity sec
2
tan
2
β
=
1
+
β. Then,
Ω
0
1/2
|
|
A
=
L
2
2
tan
2
+Ω
β
+
2σ
0
Ω+
σ
(
β
1/2
Ω+
σ
0
)
2
2
tan
2
=
+Ω
=
(
Ω+
σ
0
) sec (β
−
γ),
(4.89)
using equation (4.86). We can thus write
tan
2
(β
−
γ)
sec
2
(β
−
γ)
=
−
1
Ω+
σ
0
)
2
2
tan
2
(
+Ω
β
=
−
1
Ω+
σ
0
)
2
(
2
Ω
Ω+
σ
0
)
2
tan
2
=
β.
(4.90)
(
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