Pendule de Foucault

Choose an example or set latitude, length and initial angle, then calculate. City latitudes are rounded examples. Both views use the same physical time, initially zero. Enter an exact time in seconds, use the 0–48 h slider, or advance by a quarter of the displayed pendulum period with +T/4. Examples retain the time probe. Clear result keeps the inputs.

Play runs at real time, 1×: oscillation and precession advance together, with no separate precession speedup. Jump with the time probe to examine hours of precession. Focusing or editing the time field, moving the slider, stepping, changing parameters or clearing the result pauses playback. A hidden tool or browser tab freezes time and resumes only playback that was previously running. Playback stops at 48 h; returning after navigation restores a paused result. Reloading or changing language restores valid settings and waits for Calculate.

Local x points east and y north. The top view looks down on the horizontal plane, with positive angles counterclockwise. The pendulum is released from rest at (A, 0), where A = L sin θ. The red plane line has a marked positive end, so its reported angle and full-turn period refer to 360°. An unmarked line already overlaps itself after 180°. The trace covers at most the last two oscillations, using 97 points.

With g = 9.81 m/s², Earth rotation ΩE = 7.2921159×10⁻⁵ rad/s, f = ΩE sin φ and ω₀ = √(g/L), the linear model is x″ = −ω₀²x + 2fy′ and y″ = −ω₀²y − 2fx′. Its solution for this release condition is z = x + iy = A e^(−ift)[cos(νt) + i(f/ν)sin(νt)], with ν = √(ω₀² + f²). Both positions and the recent trace use that same solution.

The plane or elliptical major axis rotates through −ft. Precession is clockwise in the northern hemisphere, counterclockwise in the southern hemisphere and absent at the equator. A full turn takes 2π/|f|; no finite period exists at the equator. The displayed pendulum period is the usual small-angle value T = 2π√(L/g). Four times the length doubles T without changing the precession rate. The tiny Coriolis correction to the oscillation frequency is included in the motion but makes no practical display difference to that period.

The side view is a projection looking north. The pivot is (x, z) = (0, 0); the bob is (x, −√(L² − x² − y²)). The whole cable remains in view. Each view has equal horizontal and vertical metric scales, but their scales differ: the top view resolves the small horizontal displacement, while the side view includes the full length. Three-dimensional cable geometry is drawn, while the dynamics remain the linear small-angle approximation.

The model assumes constant gravity and keeps only the vertical Coriolis component. It omits damping, driving, finite-amplitude and additional elliptical precession, and variable gravity; it is not a complete spherical-pendulum model. At 5°, the familiar finite-amplitude period correction is about 0.05%. The fixed Earth rotation rate corresponds to about 86164.0900 s per rotation, not a civil 24-hour day, and is not a precision Earth-orientation value. Allowed inputs are latitude −90° to 90°, length 1–67 m, initial angle 0.1–5° and time 0–172800 s. All calculations stay in your browser.

Sources: Richard Fitzpatrick, Foucault Pendulum; Joe Wolfe, UNSW, Foucault pendulum derivation; USNO, Sidereal Time. The release-from-rest solution and constant Earth rotation rate used here are the stated model conventions.