Image 1 Caption: An animation of a Foucault pendulum at the North Pole showing the clockwise rotation of its plane of oscillation (i.e., showing its clockwise precession) RELATIVE to the Earth. The animation is time-lapsed: an actual precession period ath the North Pole is a sidereal day = 86164.0905 s = 1 day - 4 m + 4.0905s (on average).

Features:

1. Inertial frames are NOT in rotation relative to the observable universe---except for rotating frames converted to inertial frames using rotating frame inertial forces (see Mechanics file: frame_rotating.html).

   But note that very strong gravitational fields (like those very near black holes) may cause inertial frames to be intrinsically in rotation relative to the observable universe, but this is a tricky point for which yours truly CANNOT find a clear explication. The best so far (and it does NOT say much) is Wikipedia: Inertial frame of reference: General relativity.

2. A Foucault pendulum illustrates the effect of the rotating frame inertial force the Coriolis force when viewed from the rotating frame of the Earth---which is the opposite viewpoint of the animation in Image 1.

   The effect is the precession of the plane of the Foucault pendulum's oscillation relative to the the rotating frame of the Earth.

   The precession is caused by the torque of the Coriolis force.

   Torque is the twisting manifestation of a force.

3. Now why does the Coriolis force have this effect for the Foucault pendulum when we do NOT usually see this effect for most pendulums?

   The pivot the Foucault pendulum is frictionless, and so the pivot CANNOT exert any torque on the Foucault pendulum. Even a relatively small torque by friction would overcome that of the Coriolis force which is rather weak in this case. Of course, any rigid-direction pivot would completely stop the precession.

   Another reason for NOT seeing precession is that even for a Foucault pendulum, the precession period is rather long. The formula for precession period is

     P = ( 1 sidereal day )/sin(L) where L is latitude.
   
       = 1 sidereal day at L = 90°  .
   
       = [sqrt(3)/2] sdays = (0.8660 ...)  sidereal days at L = 60°  .
   
       = sqrt(2) sdays = (1.4142 ...) sidereal days at L = 45°  .
   
       = 2 sidereal days at L = 30°  .
   
       = ∞ sidereal days at L = 0°  .  

   (Wikipedia: Foucault pendulum: Examples of precession periods). Note a Foucault pendulum needs some kind of driver to keep it oscillating, but a driver that exerts NO torque.

   In the Northern Hemisphere (Southern Hemisphere), the precession clockwise (counterclockwise) relative to the rotating frame of the Earth (Wikipedia: Foucault pendulum: Mechanism).

   The simplest locations for a Foucault pendulum are at the poles. There a Foucault pendulum oscillates in a plane fixed relative to the observable universe (which is what is shown in the animation in Image 1) which means the plane of oscillation precesses relative to the Earth once per sidereal day.

   The second simplest location is the equator where the plane of oscillation does NOT precess at all relative to the Earth.

4. The Foucault pendulum is, in fact, a way to demonstrate that Earth surface locations are NOT exactly intrinsic inertial frames on the small size scale.

   Large size scale non-inertial frame effects are evidenced by weather (particularly anticyclones and cyclones: see Mechanics file: coriolis_force.html) and long-range artillery ballistics.

   

5. Image 2 Caption: An animation of illustrating a Foucault pendulum located in the dome of the Pantheon in Paris. This Foucault pendulum (although maybe NOT Foucault's Foucault pendulum: see below) has length 67 meters and is released at a distance of 50.25 meters (3/4 times its length) in the east with a zero speed. The Foucault pendulum is precessing clockwise RELATIVE to the Pantheon. The precession is time-lapsed to 110 seconds. Alas, the original caption is rather incomplete.

6. Leon Foucault (1819--1868) used a Foucault pendulum (which he did NOT invent) to demonstrate the Earth's rotation relative to the fixed stars (or as we would say now the observable universe) in 1851 (see Wikipedia: Leon Foucault: Middle years; Wikipedia: Pantheon: Under Louis Philippe I, the Second Republic and Napoleon III (1830-1871)). The Foucault pendulum was a huge one in the dome of the Pantheon in Paris.

Images:

1. Credit/Permission: Christophe_Dioux, 2023 / CC BY-SA 4.0.
   Image link: Wikimedia Commons: File:Foucault pendulum at north pole animated.gif.
   
2. Credit/Permission: User:Nbrouard, 2007 / CC BY-SA 3.0.
   Image link: Wikimedia Commons: File:Foucault-rotz.gif.