# Foucault Pendulum

## An extraordinary object

Automatic translation Updated March 12, 2022

The Foucault pendulum, named after French physicist Léon Foucault (1819-1868), is an experimental device designed to show the rotation of the Earth. The first educational experiments entitled "Come and see the earth turn" were carried out at the Panthéon in 1851.
Today, Foucault's pendulum is a 28 kg lead and brass ball attached to the dome of the Panthéon (Paris) by a 67 m long cable. A magnetic mechanism maintains its inertia movement which, due to air friction, would only oscillate for 6 hours. Its swaying makes it possible to see the terrestrial landmarks rotating (ground, walls, dome of the Pantheon, etc.). In other words, we can see the Earth rotating without even looking at celestial objects (Sun, Moon, stars). The observer rotates with the Earth and remains fixed relative to the ground. For him it is the axis of oscillation of the pendulum which turns.
This simple ordinary object forces us to accept as true several extraordinary notions.
- Once released, the Foucault pendulum oscillates along an axis which will keep the same direction indefinitely. This plane of oscillation will remain invariably fixed in space, over time, regardless of the direction in which it was launched.
- In Paris, the pendulum plane does not return to its starting point in 24 hours as one might think. At each oscillation of 16.42 seconds, it moves 5.4 mm clockwise (in the opposite direction to the rotation of the Earth) and returns to its starting point in 31h 48'. It is not the plane of the pendulum which turns but the Earth.
- The plane of the pendulum makes a complete revolution in 23 hours 56 min (sidereal day) only when it is placed at the geographical poles (North or South), because at the geographical poles, the vertical of the pendulum is parallel to the axis of rotation of the Earth.

- The closer the latitude gets to the Earth's equator, the longer the pendulums take to return to their initial axis.
- At the equator the pendulum ends up oscillating in an apparently fixed plane which does not allow to see the rotation of the Earth. This time, the vertical of the pendulum is perpendicular to the axis of rotation of the Earth. Thus, contrary to what happens at the poles, the ground does not rotate around the axis of the pendulum but carries the axis with it. For an observer everything happens as if the pendulum were transported in a train and swung in the direction of the movement of the train. The plane of the pendulum is frozen and the Earth no longer revolves around it. Its period of revolution tends towards infinity.
- In southern latitudes, the pendulum will again show the Earth's rotation and its plane will rotate counter-clockwise.
- It is extremely difficult to imagine the oscillation of a pendulum outside the poles.
All these notions are explained by a long mathematical development exposing the equations of motion of the pendulum.
The period of oscillation of the Foucault pendulum of the Pantheon is equal to 16.42 seconds because the length of the wire is equal to 67 meters. The mass of the pendulum does not matter, the length of the wire is enough to calculate the period of oscillation T.
T = 2π√l/g where l = length of wire, g = acceleration due to gravity 9.81 m/s2
At latitude θ given and an angular speed of rotation of the Earth Ω, the period of rotation is inversely proportional to the sine of this latitude, i.e. 2 π/Ωsin(θ). The sine of 30° being equal to 1/2, a Foucault pendulum implanted at a latitude of 30° will make a complete turn in 48 hours. It is the Coriolis force, perpendicular to the displacement and proportional to the speed of the pendulum, which causes the pendulum to deviate from its initial plane of oscillation.

## The pendulum is fixed relative to what?

At the time of Foucault there existed an absolute space in relation to which all movements are defined. This immutable space was therefore a natural frame of reference for the oscillation of the pendulum.
But today space or rather Einstein's space-time is a dynamic entity and the theory of relativity postulates that there is no privileged frame of reference. In the universe, absolute movement does not exist, it is always relative to another reference point which is also in motion. However, we see that Foucault's pendulum favors a precise frame of reference since its plane indicates a direction. But then, relative to what is the plane of the pendulum fixed?
This unsolved riddle is still controversial.
At the North Pole, a Foucault pendulum suspended 67 m high and launched in any direction oscillates in 16.42 s. With each oscillation, its plane deviates by 7 mm. If we throw the pendulum in the direction of the Sun, it does not seem to deviate from the Sun. But after a few hours we observe a deviation of the plane of the pendulum because the Earth/Sun direction is not fixed. Indeed, the Earth revolves around the Sun in 365 days and therefore the deviation of 7 mm, 365 times smaller, ends up appearing.
If not relative to the Sun then relative to what is the plane of the pendulum fixed?
If we throw the pendulum in the direction of any star in our Galaxy, it does not seem to deviate from the star. The distant stars seem to be the referential with respect to which the plane of oscillation of the pendulum appears to be fixed. But after a few thousand years we would observe a deviation of the plane of the pendulum because the Sun/Star direction is not fixed.

Indeed, the Sun revolves around the Galaxy in 250 million years but the star also revolves around the Galaxy and the two rotations are not synchronous. Then the deviation of the plane of oscillation with respect to the star will eventually appear.
If not relative to the stars then relative to what is the pendulum plane fixed?
It would be the same if we oriented the plane in the direction of a very distant galaxy. Drift time increases with distance from the reference object. All the landmarks will eventually come out of the pendulum plane, but then which landmark would remain in the pendulum plane?
At 13.77 billion years the drift seems to stop and the plane of the pendulum remains fixed with respect to objects close to the Big Bang. The direction of Foucault's pendulum once launched is not linked to the movement of our planet, our Sun, our galaxy, distant clusters of galaxies but to the movement of the entire observable Universe.
The axis of the pendulum is fixed then in an immutable way on this point of reference!!!
Is the pendulum sensitive to all space-time?
Is the pendulum sensitive to all objects in the universe?
170 years after its invention, the movement of Foucault's pendulum still remains mysterious and unexplained. This seemingly insignificant mechanical object surprisingly transports us to the confines of the observable Universe.

 nota: In mathematics, a conjecture is an assertion for which we do not yet know a proof, but which seems true.

### Image: The cosmic microwave background is electromagnetic radiation located in the very homogeneous microwave range observed in all directions of the sky, throughout the Universe. These slight temperature fluctuations would be the reflection of the first density variation at the origin of the clusters of galaxies. Credit: European Space Agency

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