Foucault pendulum

Choose an example or set latitude, length and initial angle. The model and rest preview update immediately. Both views share one physical time. Enter seconds, use the 0–48 h slider, or advance by a quarter of the displayed period with +T/4. Reset keeps the inputs and returns to time zero.

Play and Pause control the shared clock. Animation chooses 1×, 60× or 3600× time compression. At 1× you can observe individual oscillations; faster rates reveal precession. All positions use the same physical model and time, but rapid oscillations may be unresolved or appear to jump at high rates. The trace still covers the last two physical oscillations. Editing a parameter prepares a new rest preview. Hiding the tool pauses it; returning does not automatically resume. Only raw model inputs, example and animation rate are saved, including unfinished drafts. Reloading, changing language or returning to the tool starts a fresh comparison at zero; no elapsed time or computed motion is restored.

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.

At 48 hours, Replay starts the same calculated experiment again at time zero.