Details of the Foucault Pendulum (for the advanced student)

In the following discussion we examine the assertion by Foucault about the "fixity of the plane of oscillation" of the pendulum. The circumstances in which this
statement is exactly correct are approached by the large museum pieces such as Foucault's 67 m pendulum at the Pantheon in Paris. On the other hand, a short pendulum such as the one on display in the MacNaughton Building must rely on compensatory mechanisms to eliminate spurious turning of the plane of oscillation by non-Coriolis effects. These are described below and illustrated in the mechanical drawing of the key elements of the pendulum.

The effect (again)

To begin we inquire as to the angle turned by the pendulum bob in say l/2 of the period T of pendulum motion. For the case of a pendulum released from rest this is
the (greatly exaggerated) angle theta shown in Fig. 2 and as can be seen in that diagram the turning is the net effect of two forces, namely the Coriolis force and the
restoring force acting on the bob tending to bring it to its equilibrium position. It is a remarkable fact that if the restoring force is exactly harmonic,

Fig. 2
i.e. proportional to the displacement of the bob, the angle q = (½ T )we' which is exactly the angle through which the room has turned in the ½ period! Said
another way, the only turning of the plane of oscillation is the turning of the room and so Foucault guessed correctly that the pendulum support would have no effect since it also turns with the room. Furthermore, since this precession or turning of the plane of oscillation is proportional to we' it does indeed measure the sine of the latitude as Foucault said.

One can give an argument which should make this result somewhat less of a surprise. When the pendulum is viewed from the frame of reference of the space of the
fixed stars there is no Coriolis force and the motion is that of an harmonic oscillator. One then knows the motion can be described in time by x = a cos(t),
y = bsin( t) which is the equation of an ellipse with semi-major and minor axes a and b and where the frequency w = 2p/T. The ellipse is shown in Fig. 3 (also
exaggerated) and since one is describing the same motion as in Fig. 2 but from a different point of view, there is a simple correspondence between points on the
path of motion. In particular the ellipse semi-axes values a and b are as given in Fig. 2 as the distances of furthest and closest approach to the equilibrium point.
Finally, since the ellipse remains forever fixed in orientation in the reference frame of the stars it must be that the only rotation is the relative rotation of the earth. This
fixes the angle theta in Fig. 2 as precisely we' times the time taken to get from A to C, namely ½ T.

Fig. 3

The correct scale, b = we' a, for the quantities shown in Fig. 2 follows from dy/dt = b  cos(t) and the fact that the point of release of the bob at t = 0,
although fixed in the room, is moving at speed we' a in the star frame. With a approximately 6 cm and w approximately 3.5 sec-1 for our pendulum, b approximately
equals 10-3 mm, hardly a perceptible value!