Environmental Engineering Reference
In-Depth Information
w
N
e
2
e
3
e
r
e
1
x
l
Figure 8.4
A non-inertial system of co-ordinates defined at rest relative to the Earth.
Although we have drawn the basis vectors on the surface of the Earth we should
remember that the basis vectors define only directions in space. To complete our
specification of the co-ordinate system we need also to specify the location of the
origin and it is convenient to choose the origin to be located at the centre of the
Earth. In this case the point on the surface of the Earth at which we have drawn
the basis vectors is located at position
x
=
R
e
3
(8.24)
and the angular velocity of the Earth is
ω
=
ω(
cos
λ
e
2
+
sin
λ
e
3
).
(8.25)
We are now ready to explore the influence of the Earth's spin upon physics
occurring in the vicinity of
x
, which is a general point on the Earth's surface.
We start by computing the centrifugal force acting upon a particle located at
x
.
Of course we already know the answer from our prior understanding of circular
motion: it should be a force of magnitude
mω
2
R
cos
λ
pointing in the
e
r
direction
(see Figure 8.4). To warm up, let us compute it using Eq. (8.23). First we compute
the vector
ω
×
x
:
ω
×
x
=
ωR
cos
λ
e
1
(8.26)
and since
ω
×
e
1
=−
ω
e
r
it follows that the centrifugal force is
mω
2
R
cos
λ
e
r
,
−
ω
×
ω
×
=
m
(
r
)
(8.27)
