Watch the Earth turn
Léon Foucault hung a pendulum in Paris in 1851, set it swinging in a straight line, and let people watch the plane of its swing slowly rotate — the first time anyone had seen the Earth spin without looking at the sky. The rate it turns is Earth's own spin times one factor: the sine of your latitude. Drag the latitude below and watch the plane precess. Then watch the very same number fall out of pure geometry, with no forces at all.
There is a subtlety almost every explanation gets wrong, and this page is built around getting it right: the plane does not, in general, "stay fixed while the Earth turns under it." That is true only at the poles. Everywhere else the honest statement takes two numbers that add up to one turn of the Earth — and you can watch both.
1 · The pendulum, seen from above
Below is the view straight down onto the pendulum, the way the great installations lay it out — the swinging plane leaves a slow rose as it turns, and in Foucault's own halls a ring of pins gets knocked over one by one to prove the rotation is real. The compass (N·E·S·W) is fixed to the room — to the floor you stand on. Drag your latitude and set the clock speed; the swing plane rotates at exactly Ω·sin φ, where Ω is Earth's spin.
| your latitude φ | 48.9° |
| precession rate Ω·sinφ | 11.33°/hr |
| direction (N hemisphere) | clockwise ↻ |
| one full turn takes | 31.8 h |
| = sidereal day ÷ sinφ | 23.93 / 0.753 |
| plane turned so far | 0.0° |
The back‑and‑forth is shown at a watchable cadence; the rotation of the plane against the room is the real physics — Ω·sinφ per unit of the clock at top right. At the equator (drag φ to 0.5°) the rose freezes: no turn. At the pole (φ = 90°) it makes a full turn in one sidereal day, 23 h 56 m.
Where does the sinφ come from? Stand in the rotating frame of the Earth. A moving bob feels the Coriolis force, −2m Ω×v. Only the vertical part of Earth's spin vector — its component along your local "up" — twists the horizontal swing, and that vertical part is Ω·sinφ: full at the pole where up is the spin axis, zero at the equator where up is perpendicular to it. Write the small‑swing equations with z = x + iy and they read
That leading e−i(Ω sinφ)t is the whole story: it takes the ordinary back‑and‑forth and rotates it, clockwise in the north, at rate Ω sinφ — and crucially independent of the pendulum's length or mass. The bob's own frequency ω₀ only sets how fast it swings, never how fast the plane turns. The animation above renders this exact solution; the offline check integrates the raw equation from scratch and recovers the same rate to four figures.
2 · The same turn, without any force
Here is the part worth the trip. Forget the Coriolis force entirely. Think only about geometry. The swing direction is an arrow lying flat on the ground, and as the Earth carries the pendulum eastward it drags that arrow around a circle of latitude on the sphere — carrying it as "straight ahead" as a curved surface allows, which mathematicians call parallel transport. Because a latitude circle (except the equator) is not a great circle, the arrow does not come home pointing the way it left. The mismatch is the precession.
| turn vs. the ground | 271.1° |
| = 2π·sinφ (the precession) | 271.1° |
| cap solid angle | 88.9° |
| = 2π·(1−sinφ) vs. stars | 88.9° |
The gold arrow is the swing direction, parallel‑transported. The faint grey arrow marks where it started. Watch the gap open as it rounds the circle: after one full lap the gap is the day's precession, 2π·sinφ — read against the ground. The blue cap above your latitude has area (solid angle) 2π(1−sinφ): that is the very same plane's rotation measured against the fixed stars. The two are complements.
The reconciliation almost no page states. One rotation of the Earth is 360°. The Earth splits that single turn between the two frames, by latitude:
2π·sinφ (Foucault)
2π·(1−sinφ) (solid angle)
always, every latitude
Read it at the extremes and it clicks. At the North Pole (φ = 90°): the floor‑turn is the whole 360° — the plane sweeps a full circle against the ground each day — while the star‑turn is 0°, the plane genuinely hangs fixed among the stars as the Earth pirouettes beneath it. That pole case is the one everybody pictures, and it is exactly where the popular "it stays fixed in space" line is true. At the equator (φ = 0°): the floor‑turn is 0° — no precession, the rose never moves — while the star‑turn is the full 360°, the plane simply riding around once a day with the ground it lies on. Every latitude in between shares the turn in the ratio sinφ : (1−sinφ). This "leftover" rotation is a geometric phase — an angle you accumulate not from any force but purely from the shape of the loop you were carried around; the mathematics is identical to a falling cat righting itself and was recognised as such by M. V. Berry and J. H. Hannay in the 1980s.
3 · What Foucault actually did — and what fights the experiment
On 3 February 1851, in the Meridian Room of the Paris Observatory, Foucault posted an invitation that has never been bettered as a piece of scientific theatre: « Vous êtes invité à venir voir tourner la Terre » — "You are invited to come and see the Earth turn." Weeks later, in late March 1851, he hung the famous version under the dome of the Panthéon: a 28‑kilogram brass‑clad lead bob on a 67‑metre wire (that length gives a stately 16.4‑second swing). At the Panthéon's latitude, 48.9°N, the plane crept clockwise about 11.3° every hour, closing a full circle in roughly 31.8 hours. It was the first bench demonstration of a fact until then known only from the sky; the next year Foucault built a second proof and, in the naming, gave us the word gyroscope.
The honest apparatus note. A real bob almost never swings in a perfect plane. The faintest sideways nudge at release sends it around a thin ellipse instead of a line — and a swinging ellipse, at finite amplitude, precesses on its own (this is Airy precession, in the same sense as the swing), an effect that can swamp or bias the true Earth‑turn. The classic fix is a burnt‑thread release: draw the bob aside, tie it off, let it come dead still, then burn the thread so it leaves with zero sideways push. Even that only tames the launch — residual ellipticity from an imperfect, non‑symmetric suspension keeps leaking in, which is why modern pendulums add correction rings and drives. The clean Ω·sinφ is the truth the apparatus is always fighting to show.
Show the check
Every number on this page — the Ω·sinφ precession, the 11.3°/hr and 31.8 h at Paris, the parallel‑transport turn and its solid‑angle complement, and the fact that they sum to one Earth turn — is recomputed offline, from scratch, in research/watch-the-earth-turn/verify.mjs (30 checks, all passing). The rate is confirmed three independent ways that must agree:
- The closed form. The precession factor e−i(Ω sinφ)t is shown to satisfy the swing equation exactly, and to be independent of the pendulum's own frequency.
- The simulation. The raw rotating‑frame equation is RK4‑integrated with no assumed answer; the swing line's drift is measured and comes out Ω·sinφ to four figures, at several latitudes and two pendulum lengths.
- The geometry. A tangent vector is parallel‑transported around the latitude circle on the unit sphere; against the ground frame its net turn is 2π·sinφ, and the complement to a full turn matches the cap's solid angle 2π(1−sinφ) to machine precision — the equator, being a geodesic, gives zero.
Run it yourself from a clean checkout: node research/watch-the-earth-turn/verify.mjs. Historical specifics (the 1851 dates, the 67 m / 28 kg Panthéon figures, the invitation, the Airy‑precession caveat, and the Berry–Hannay geometric‑phase framing) were cross‑checked against MacTutor, Britannica, the Panthéon's own record, and the 2025 Am. J. Phys. treatment of elliptical vs. Coriolis precession; where sources genuinely disagree — the exact Panthéon day (26 vs. 31 March) — the page says "late March" rather than pick one.