Environmental Engineering Reference
In-Depth Information
as we anticipated. Phenomenologically, the centrifugal force has the effect of
slightly reducing the weight of objects on the surface of the Earth, the effect being
greatest at the equator where cos
λ
1. Note also that the centrifugal force does
not acts downwards, rather it acts radially outwards from the axis of the Earth's
rotation. This means that a pendulum suspended above the Earth's surface will not
point exactly towards the centre of the Earth.
=
Example 8.2.2
Compute the maximum deflection of a pendulum suspended close
to the surface of the Earth.
Solution 8.2.2
The net force on a particle suspended close to the Earth's surface
at a latitude λ is given by the sum of the gravitational force (i.e. the weight) and
the centrifugal force:
F
=
−
ω
×
ω
×
m
g
m
(
r
).
We already deduced the centrifugal force, but to determine the deflection of a pen-
dulum we should express the vector
e
r
in terms of our basis vectors, i.e.
e
r
=−
sin
λ
e
2
+
cos
λ
e
3
(8.28)
so that
F
=−
ω
2
R
cos
λ
sin
λ
e
2
+
(ω
2
R
cos
2
λ
−
g)
e
3
since
g
g
e
3
.
Now for a pendulum at rest this net force is balanced by the tension in the pen-
dulum, and so the pendulum aligns itself with this force and it is thus deflected at
an angle α to the vertical. This deflection is illustration in Figure 8.5 in the case of
a pendulum hanging in the northern hemisphere and is given by
=−
ω
2
R
cos
λ
sin
λ
g
tan
α
=
ω
2
R
cos
2
λ
.
−
45
◦
(you should convince yourself that this
is indeed the case: note there is no deflection at the poles or on the equator). Since
The maximum deflection occurs at λ
=
Northern
hemisphere
a
N
g
eff
Figure 8.5
The deflection
α
of a pendulum suspended close to the surface of the Earth.
