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Consider a pendulum consisting of a compact mass suspended from a light cable of length in such
a manner that the pendulum is free to oscillate in any plane whose
normal is parallel to the Earth's surface. The mass is
subject to three forces: first, the force of gravity , which
is directed vertically downward (we are again ignoring centrifugal
acceleration); second, the tension in the cable, which is directed upward
along the cable; and, third, the Coriolis force. It follows that the
apparent equation of motion of the mass, in a frame of
reference which corotates with the Earth, is [see Equation (414)]

(438) 
Let us define our usual Cartesian coordinates (,,), and
let the origin of our coordinate system correspond to the equilibrium position
of the mass. If the pendulum cable is deflected from the downward vertical by a small angle
then it is easily seen that
,
,
and
. In other words, the change in height
of the mass, , is negligible compared to its horizontal displacement.
Hence, we can write , provided that . The
tension has the vertical component
,
and the horizontal component
, since
see Figure 27. Hence, the
Cartesian equations of motion of the mass are written [cf., Equations (424)(426)]
To lowest order in (i.e., neglecting ), the final equation, which is just vertical force balance, yields .
Hence, Equations (439) and (440) reduce to
Figure 27:
The Foucault pendulum.

Let

(444) 
Equations (442) and (443) can be combined to give a single complex
equation for :

(445) 
Let us look for a sinusoidally oscillating solution of the form

(446) 
Here, is the (real) angular frequency of oscillation, and
is an arbitrary complex constant. Equations (445) and (446)
yield the following quadratic equation for :

(447) 
The solutions are approximately

(448) 
where we have neglected terms involving
.
Hence, the general solution of (446) takes the form

(449) 
where and are two arbitrary complex constants.
Making the specific choice , where is real, the
above solution reduces to

(450) 
Now, it is clear from Equation (444) that and are the real and imaginary
parts of , respectively. Thus, it follows from the above that
These equations describe sinusoidal oscillations, in a plane whose normal
is parallel to the Earth's surface, at the standard pendulum frequency .
The Coriolis force, however, causes the plane of oscillation to slowly precess at the
angular frequency
. The period of the
precession is

(453) 
For example, according to the above equations, the pendulum oscillates
in the direction (i.e., north/south) at , in the direction
(i.e., east/west) at , in the direction again at , etc. The precession is clockwise (looking from above)
in the northern hemisphere, and counterclockwise
in the southern
hemisphere.
The precession of the plane of oscillation of a pendulum, due to
the Coriolis force, is used in many museums and observatories to
demonstrate that the Earth is rotating. This method of making the
Earth's rotation manifest was first devised by Foucault in 1851.
Next: Exercises
Up: Rotating Reference Frames
Previous: Coriolis Force
Richard Fitzpatrick
20110331