How Does a Foucault Pendulum Prove Earth Rotates?

You do not need a satellite to see Earth turn. In 1851 Léon Foucault hung a long pendulum in Paris and watched its swing plane drift clockwise across the floor. The room was rotating under the pendulum. Our sticker Pendulux is that demonstration in pixel form.

What did Foucault do in 1851?

Foucault’s public demonstration at the Panthéon used a heavy bob on a long wire, started carefully so it swung in a fixed plane relative to the inertial sky. Relative to the building, that plane appeared to rotate. Encyclopaedia Britannica’s article (archived copy) recounts the experiment and the latitude dependence (Britannica, Foucault pendulum^[1]). Paris Observatory still presents the pendulum as a house classic (Paris Observatory^[2]).

The point was philosophical as much as technical: Earth’s rotation became a laboratory fact, not only an astronomical inference from the stars.

Why does the swing plane seem to turn?

In an inertial frame the pendulum’s swing plane stays fixed (ignoring tiny dissipation and the Earth’s orbit). The Earth rotates under it. In the rotating lab frame you describe the same motion with Coriolis forces that slowly twist the apparent plane of oscillation.

At the North Pole the floor turns a full 360° under the pendulum in one sidereal day, so the swing plane appears to rotate once per sidereal day. At the Equator the local vertical is perpendicular to Earth’s spin axis in such a way that the Coriolis effect does not rotate the swing plane at all. Between those extremes the rate scales with the sine of the latitude.

What is the precession rate formula?

Britannica states the rule in words: the rate of rotation of a Foucault pendulum equals the Earth’s rotation rate times the sine of the latitude (Britannica^[1]). Per sidereal day that is

Ω_F = 360° · sin(φ) per sidereal day

or about 15° · sin(φ) per hour, with φ the geographic latitude. Cross-checks of the same formula appear in standard references (Wikipedia, Foucault pendulum^[3]).

• Pole (φ = 90°): sin φ = 1 → ~360° per sidereal day (~15°/h).
• Latitude 30°: sin 30° = 1/2 → ~7.5°/h (Britannica’s Cairo/New Orleans example).
• Equator (φ = 0°): sin φ = 0 → no rotation of the plane.
• Paris (~48.9° N): sin φ ≈ 0.75 → roughly 11°/h, a period near 32 hours — matching historical accounts of Foucault’s installation (Britannica^[1]).

Use a sidereal day (~23 h 56 min), not a solar day, if you want the astronomy-grade number; the difference is small for a classroom demo.

What does this have to do with time’s arrow?

Nothing directly — Earth’s rotation is a periodic clock, not entropy. We filed this topic under the time pillar because Foucault turned an abstract astronomical period into a visible indoor clock. For entropy and the thermodynamic arrow see Why Does Time Only Go Forward? and What Exactly Is Entropy?.

Wear the swing

Pendulux lives with the other law stickers in Physicist Stickers / law-character stickers, and the wider sky of Relativity & Cosmos.

How do you build a Foucault pendulum that works?

You need a long wire (tens of metres help), a heavy bob, a bearing that does not prefer one azimuth, and a clean start with no sideways push. Short desk pendulums are dominated by elasticity and ellipticity of the swing; they will not show a clean Foucault drift. Museum installations still use the same idea Foucault used in 1851 (Britannica^[1]; Paris Observatory^[2]).

Deriving the sine factor (sketch)

Earth’s angular velocity vector Ω points along the spin axis. At latitude φ the local vertical makes an angle (90°−φ) with that axis, so the component of Ω along the local vertical is Ω sin φ. That vertical component is what twists the horizontal swing plane. Hence

Ω_F = Ω_Earth · sin φ = 360° · sin φ per sidereal day.

Britannica’s numerical examples (15°/h at the pole, 7.5°/h at 30°, ~11°/h at Paris) match that formula (Britannica^[1]). Independent write-ups quote the same Ω ∝ sin φ law (Wikipedia^[3]).

What Foucault did not claim

The pendulum does not measure the length of the year, the shape of Earth’s orbit, or quantum gravity. It measures the projection of Earth’s spin on the local vertical. That is already enough: it turned a global astronomical fact into a continuous indoor demonstration anyone could watch.

Classical mechanics, worn thin

Pendulux sits with other classical-law stickers. If you want the thermodynamic arrow rather than the rotational clock, jump to Why Does Time Only Go Forward?.

Classroom physics papers on the same effect

Physics-teacher journals keep rediscovering clean ways to see Coriolis in the Foucault geometry. Edge’s short note ties the horizontal Coriolis acceleration to the observed drift (Edge 1982^[4]). Olson’s “pilgrimage” piece revisits the Paris installations for instructors (Olson 2021^[5]). Wilson’s 2024 note again applies the Coriolis effect explicitly to the pendulum (Wilson 2024^[6]). None of them changes Foucault’s 1851 result; they repackage the sine-of-latitude law for teaching.

Together these teaching notes and Britannica’s latitude examples are enough to trust the formula Ω_F = 360°·sin(φ) per sidereal day without inventing a new derivation.

FAQ

Does the pendulum prove Earth is round?

No. It proves Earth rotates. Roundness is a separate set of measurements.

Why sidereal day, not 24 hours?

A sidereal day is one rotation relative to the distant stars — the inertial reference the swing plane keeps. A solar day is about four minutes longer because Earth also orbits the Sun.

Which way does it turn?

Clockwise when viewed from above in the Northern Hemisphere, counterclockwise in the Southern Hemisphere.

References

1. Encyclopaedia Britannica, “Foucault pendulum” (archived 2023-12-31). https://web.archive.org/web/20231231045523/https://www.britannica.com/science/Foucault-pendulum
2. Paris Observatory (PSL), “Foucault’s pendulum”. https://observatoiredeparis.psl.eu/le-pendule-de-foucault.html?lang=en
3. Wikipedia, “Foucault pendulum” (formula cross-check). https://en.wikipedia.org/wiki/Foucault_pendulum
4. R. D. Edge (1982), “Foucault pendulum and the Coriolis effect,” Phys. Teach. 20, 116. https://doi.org/10.1119/1.2340964
5. D. W. Olson (2021), “A Foucault Pendulum Pilgrimage in Paris,” Phys. Teach. 59, 232. https://doi.org/10.1119/10.0004143
6. A. E. Wilson (2024), “Applying the Coriolis Effect to the Foucault Pendulum,” Phys. Teach. 62, 508. https://doi.org/10.1119/5.0150867

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.