Foucault Pendulum

The setup

A heavy bob hangs from a long, freely pivoting wire. Set it swinging in a single vertical plane. Wait. Hours later the swing plane has rotated, neatly tracing a series of arcs in the sand on the floor below. The original 1851 demonstration used a 28 kg bob on 67 m of piano wire under the Panthéon dome, with a delicate pin on the bob to score a circle of sand. Every cycle the pin moved a few millimetres along the arc. Onlookers had to wait minutes between visible rotations.

Precession period and rate

The swing plane precesses at angular rate Ω sin(φ), where Ω = 2π / T_sidereal = 7.292 × 10⁻⁵ rad/s = 15.04°/h is Earth's rotation rate (one sidereal day = 23 h 56 min 4.1 s), and φ is the local latitude.

T_precess = T_sidereal / sin(φ) = 23.93 h / sin(φ)

Location	Latitude φ	sin(φ)	Precession rate	Period
North Pole	90.00°	1.000	15.04°/h	23.93 h
Saint Petersburg	59.93°	0.865	13.01°/h	27.67 h
London	51.50°	0.783	11.77°/h	30.58 h
Paris (Panthéon)	48.85°	0.753	11.32°/h	31.79 h
New York	40.71°	0.652	9.80°/h	36.71 h
Singapore	1.35°	0.024	0.36°/h	1015 h (42 d)
Equator	0.00°	0.000	0°/h	∞
Buenos Aires	−34.60°	−0.568	−8.54°/h (reverse)	42.15 h

In the Southern Hemisphere the swing plane rotates in the opposite sense (counter-clockwise viewed from above) because sin(φ) flips sign.

The inertial-frame explanation

Step off the spinning Earth into an imaginary frame that does not rotate (an inertial frame). In that frame, Newton's laws are simplest: the only horizontal forces on the bob are gravity's tangential component, which restores the bob to its vertical — and this force lies in the plane of the swing. So the swing plane is fixed in inertial space; it does not rotate at all. What rotates is the building beneath the pendulum, carried around once per sidereal day with the rest of Earth's surface.

An observer fixed to the building sees their floor as stationary, so they instead see the pendulum's plane drift backwards relative to them. The rate at which their local "up" twists about the swing plane is the vertical component of Earth's spin: Ω sin(φ). Below the equator that vertical component points down rather than up, so the sense of precession reverses.

The rotating-frame explanation

If you insist on staying in the rotating frame, you have to include the Coriolis pseudo-force F_C = −2 m Ω × v. Decompose Ω into a vertical part Ω sin(φ) ẑ and a horizontal part Ω cos(φ) along the local meridian. The horizontal piece gives a vertical Coriolis kick that averages to zero over a full swing (up on the outbound, down on the return). The vertical piece gives a horizontal Coriolis force perpendicular to v, which steers the bob along a slowly rotating ellipse — exactly the Foucault precession at rate Ω sin(φ).

Worked example — Panthéon parameters

The 1851 reconstruction uses L = 67 m, m = 28 kg, initial amplitude θ₀ = 6° (≈ 7.0 m sideways at the bob).

Quantity	Value
Swing period T = 2π√(L/g)	16.42 s
Swings per hour	219
Precession rate (φ = 48.85°)	11.32°/h = 0.0316°/swing
Sideways arc movement per swing (at 6° amplitude)	≈ 3.9 mm at the bob
Time to rotate 360°	31.79 h

The original 1851 demonstrations used overlay rings of sand at the bob — the pin on the bob's underside tipped over the next ring each cycle, leaving a slowly rotating star pattern of disturbed sand.

The rosette pattern

In practice the bob does not return exactly to a point on each swing — small initial sideways velocity, air currents, or pivot asymmetry make the bob trace an ellipse rather than a line. Over many cycles, that ellipse itself rotates, and the trace on the sand becomes a rosette: a sequence of overlapping arcs whose envelope is the precession.

Elliptical motion introduces a parasitic precession proportional to the ratio of minor to major axes, called the Airy precession. To suppress it, Foucault installations:

• Release the bob from a thread that is burned through cleanly, ensuring zero initial sideways velocity.
• Use a Charron ring — a circular metal disc just below the suspension point that absorbs any sideways energy without affecting the in-plane swing.
• Re-pump the swing electromagnetically along the current swing direction to compensate for damping.

Variants

• Bravais pendulum. A horizontal conical pendulum, run as a Foucault. The bob rotates in a circle, and the orientation of that circle precesses at the same rate Ω sin(φ).
• Gyrocompass. A spinning gyroscope mounted on a frictionless gimbal aligns its axis with Earth's rotation, settling north–south. Same physics, simpler answer — no rosette to read.
• Ring-laser gyro and atom-interferometer gyro. Modern inertial navigation uses interference of light or matter waves around a loop to measure rotation rates to 10⁻¹¹ rad/s — millions of times more sensitive than Foucault, but the principle is the same: detect rotation by comparing the round-trip phase of a wave around the loop.
• Foucault on Mars. Mars rotates in 24 h 39 min sidereal. A Mars Foucault at the polar caps would precess almost identically to Earth's polar Foucault. The 2021 Perseverance rover did not bring one, but it would have worked.
• Foucault on a turntable in lab. Spin a turntable at 1 rev/s and mount a small pendulum on it. Replicates Foucault precession with T_precess = 1 s, observable in a few seconds — a popular demonstration alternative to a six-storey installation.
• Foucault in microgravity. Aboard the ISS the pendulum has no preferred plane, since gravity does not restore it. The experiment relies on local g, so it fails in free fall — but the underlying Earth rotation still drives Coriolis effects on fluids and gyroscopes.

JavaScript — Foucault precession and rosette

const SIDEREAL_HR = 23.9344696;  // length of a sidereal day in hours
const OMEGA = 2 * Math.PI / (SIDEREAL_HR * 3600);  // rad/s, Earth rotation

function foucaultPeriod(latitudeDeg) {
  const sinPhi = Math.sin(latitudeDeg * Math.PI / 180);
  return SIDEREAL_HR / Math.abs(sinPhi);
}

function foucaultRate(latitudeDeg) {
  return OMEGA * Math.sin(latitudeDeg * Math.PI / 180);  // rad/s
}

// Panthéon
console.log(`Paris: ${foucaultPeriod(48.85).toFixed(2)} h`);   // 31.79
console.log(`London: ${foucaultPeriod(51.5).toFixed(2)} h`);   // 30.58
console.log(`North Pole: ${foucaultPeriod(89.9).toFixed(2)} h`); // 23.94
console.log(`Equator: ${foucaultPeriod(0.001)} h`);            // Infinity

// Bob trajectory in the ground frame
function foucaultTrajectory(L, theta0, latitudeDeg, dt, T_total) {
  const g = 9.81;
  const omega_s = Math.sqrt(g / L);             // swing angular frequency
  const omega_p = foucaultRate(latitudeDeg);    // precession rate (rad/s)
  const path = [];
  for (let t = 0; t <= T_total; t += dt) {
    const r = theta0 * L * Math.cos(omega_s * t);
    const psi = omega_p * t;                    // current swing plane azimuth
    path.push({ x: r * Math.cos(psi), y: r * Math.sin(psi), t });
  }
  return path;
}

// Trace 4 hours of Panthéon swings at 6° amplitude, dt = 0.5 s
const trace = foucaultTrajectory(67, 6 * Math.PI / 180, 48.85, 0.5, 4 * 3600);
console.log(`Plane rotated ${(foucaultRate(48.85) * 4 * 3600 * 180 / Math.PI).toFixed(1)} ° in 4 h`);  // 45.3°

Where Foucault physics matters

• Demonstration of Earth's rotation. The first table-top proof — still the standard museum demonstration of an otherwise abstract fact.
• Inertial navigation. Submarines, aircraft, and spacecraft use ring-laser or atom-interferometer gyros that exploit the same rotational signature, accurate to centimetres per hour.
• Coriolis-driven phenomena. Trade winds, hurricane rotation sense, ocean gyres, and even the curvature of long artillery trajectories share the Ω sin(φ) factor.
• Length-of-day measurement. Combining a Foucault-style pendulum with modern timekeeping enables direct geophysical inference of changes in Earth's rotation rate.
• Geodesy and earthquakes. Large earthquakes change Earth's moment of inertia and shorten the day by microseconds. Inertial sensors detect this through subtle precession-rate changes.
• Education. A Foucault pendulum in a science museum bridges three concepts simultaneously: rotating reference frames, fictitious forces, and the inertial nature of free swinging — invaluable for visitors who have never thought about non-inertial frames.
• Eco Pro's "Day Counter." A Foucault installation can serve as a literal solar-noon clock for the latitude where it is installed: noting when the swing plane returns to a fixed direction gives the local sidereal day to within a percent.

Common mistakes

• Using a solar day (24 h) instead of a sidereal day (23.93 h). Earth's rotation rate relative to inertial space is one rotation per sidereal day, not per solar day. The 4-minute difference matters over long observations.
• Forgetting sin(latitude). At the equator nothing happens. The naïve answer "24 h everywhere" is wrong.
• Confusing precession direction with hemisphere myth. Foucault precesses clockwise in the Northern Hemisphere, counter-clockwise in the Southern — opposite of the popular (and false) toilet-water claim.
• Ignoring elliptical (Airy) precession. Sloppy release introduces sideways velocity that biases the rate. Modern installations use a thread-burn release and a Charron ring.
• Using a short pendulum. A 1 m pendulum has T_swing = 2 s, sweeping ~0.3 mm per swing at typical amplitudes — invisible against drag-induced wobble. Foucault chose 67 m specifically to make the effect dominate noise.
• Calling it the "Coriolis force in action." It is, in the rotating frame. But it is equally well a demonstration of Newton's laws in an inertial frame — the swing plane is fixed and the ground turns. The two views are equivalent.

Performance notes — energy budget

A 28-kg bob at 6° amplitude on 67 m of wire has total mechanical energy ½mv² + mgh ≈ 33 J. Air drag and pivot friction extract about 0.3–0.5 J per cycle, so without active replenishment amplitude halves in roughly 8 hours. A Charron-ring electromagnetic boost of 50 mJ per swing keeps the amplitude steady indefinitely, drawing under 5 W from a wall socket — less than a phone charger sustains the most famous physics demonstration of the 19th century.