How does a Foucault pendulum work?

The statement "the plane of the oscillation of the pendulum is fixed while the earth rotates underneath." is applicable only for a polar pendulum. On every other latitude there is more going on.

To drive that point home let me go over the following case: at the latitude of Paris it takes 32 hours for the plane of swing to go through a complete rotation. Of course that means that in 24 hours the plane of swing goes through 3/4 of a complete rotation. Hence if at t=0 the plane of swing is north-south then 24 hours later, when the Earth is back in the same orientation as 24 hours before, the plane of zwing is east-west.

The thing that has happened in those 24 hours is that the swinging pendulum bob has been exchanging momentum with the Earth. Of course, the change of momentum of the Earth is negligable, but that does not change the fact that exchange of momentum has taken place.

To make that exchange of momentum vivid I give the following thought demonstration. Imagine a small celestial body, say some asteroid, roughly spherical. Let that sphere be rotating. Build a structure such that a pendulum bob can swing, make that structure strong enough that the mass of the pendulum bob can be significant percentage of the mass of the sphere. Let's say the mass of the pendulum bob is 5% of the mass of the sphere.

With such a setup the swing of the pendulum bob will have a measurable effect on the orientation of the sphere. The sphere and the bob are exchanging momentum. That is the purpose of this thought demonstration: the same reasoning extends to a large celestial body: as a pendulum is swinging it is exchanging momentum with the celestial body.

For a pendulum not located on either of the poles that is why the plane of swing is rotating at a smaller angular velocity than the angular velocity of the celestial body itself.

Indeed it appears as if this dynamic effect cannot be strong enough. Then again, it is cumulative, that makes all the difference

Now, most derivations that you will encounter do not describe the dynamics in these terms. Yet they arrive at the actually observed rate at which the plane of swing is precessing. How does that happen?

Well, generally derivations are constructed in reverse. The construction of the derivation works backwards from the known observation.
Then in the final write-up the derivation is presented in forward steps.
The derivation then contains assumptions that produce the desired result, and these assumptions look plausible, but they are not necessarily physically justified assumptions.

Next, let me take the case of the actual Foucault pendulum in the Pantheon. Foucault describes that on rare occasions there was time for long uninterrupted runs. Over the course of such a long run the amplitude of the swing would decay to a mere 10 centimeters, but Foucault reports that the plane of swing was still rotating at the same rate.

The Foucault pendulum in the Pantheon has a 67 meter cable. At the latitude of Paris the distance to the Earth's axis is such that the required centripetal force to sustain circumnavigation of the Earth's axis corresponds to a weight on a 67 meter cable to be displaced about 10 centimeter.

Two jobs
So: the suspending cable has two jobs to do: it must provide the required centripetal force to sustain circling the Earth's axis, and it provides the restoring force for the swinging motion.

Let me now discuss 4 cases:

• the pendulum bob swings south to north:
  The centripetal force is doing work, increasing the bob's angular velocity

• the pendulum bob swings north to south:
  The centripetal force is doing negative work, decreasing the bob's angular velocity

• the pendulum bob swings west to east:
  Now the pendulum bob is circumnavigating the Earth's axis faster than the Earth itself. Because of that velocity surplus the pendulum bob will swing wide.

• the pendulum bob swings east to west:
  During that part of the swing the pendulum bob is circumnavigating the Earth's axis slower than the Earth itself. Because of that velocity deficit the pendulum bob is pulled closer to the Earth's axis

Interestingly, to a very acceptable approximation the rotation-of-Earth-effect is equally strong in every direction of the plane of swing.

Importantly, if there would not be a rotation-of-Earth-effect for east-west swing then the rotation of the plane of swing would stall there. We know it doesn't stall, hence any explanation that fails to predict/explain the effect for east-west swing is definitely wrong.

The image below gives a schematic presentation of the cumulative effect.

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