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,

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 w

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* = *b*sin( *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 w_{e}'
times the time taken to get from A to C, namely ½ T.

The correct scale, *b* = w_{e}'
a, for the quantities shown in Fig. 2 follows from d*y*/d*t *=
*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 w_{e}'
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!