Why is the precession rate proportional to sin(latitude)?

Earth's spin vector Omega points along the polar axis, but at latitude phi only the vertical component Omega*sin(phi) acts about the local vertical. Since the pendulum cares only about the vertical Omega component, the swing plane rotates at Omega*sin(phi). At 35 degrees latitude, sin(35) is about 0.574, so a full rotation takes about 41.7 hours; at the pole 23.93 hours (one sidereal day); at the equator the rate is zero.

Do pendulum length or mass affect the precession?

No. The precession rate Omega*sin(phi) depends only on latitude and is independent of length L, mass m, and amplitude. Length sets only the swing period T = 2*pi*sqrt(L/g): longer pendulums simply swing more slowly. Real Foucault pendulums use 60 to 70 meter wires with heavy bobs not for faster rotation but to delay air-drag damping so that the swing persists for many hours and the precession becomes clearly visible.

How is the Coriolis force related to the Foucault pendulum?

Viewed from the rotating Earth (a non-inertial frame) any moving object experiences the Coriolis force F = -2 m * Omega cross v. As the pendulum mass moves, this force acts perpendicular to its motion and gradually rotates the swing plane. In the northern hemisphere the deflection is to the right (clockwise rotation); in the southern hemisphere to the left (counterclockwise). The same mechanism explains hurricane spin direction and ballistic deflection.

Foucault Pendulum Simulator — Earth Rotation Precession

Q: Why does a Foucault pendulum's swing plane appear to rotate?

In an inertial frame the swing plane stays fixed in direction, but the floor (the observer) rotates with the Earth. Seen from the ground, the plane appears to rotate the opposite way: clockwise in the northern hemisphere, counterclockwise in the southern, and not at all on the equator. Leon Foucault demonstrated this in 1851 at the Pantheon in Paris with a 67 m pendulum, the first laboratory-scale visual proof of Earth rotation.

Parameters

Latitude φ

Pendulum length L

Gravity g

m/s²

Elapsed time t

The sidereal day 86164.1 s (one inertial Earth rotation) sets Omega_earth = 2*pi / T. The plane appears clockwise in the northern hemisphere and counterclockwise in the southern.

Results

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Swing period T_osc

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Full-rotation time

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Precession rate

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Precession angle at t

Top view of the swing plane (floor)

center = directly below pivot / blue solid = current swing axis / gray dashed = swing axis at t=0 / orange arc = precession angle Δθ / N/E/S/W = compass

Precession angle vs time (latitude comparison)

x = elapsed time t (h) / y = precession angle Δθ (°) / blue = current latitude / gray = equator (0°), 35°, pole (90°) / yellow dot = current (t, Δθ)

Theory & Key Formulas

The Earth's spin vector Omega contributes only the component along the local vertical at latitude phi. That single component drives the rotation of the Foucault swing plane.

Simple-pendulum swing period (small amplitude):

$$T_{\mathrm{osc}} = 2\pi\sqrt{\dfrac{L}{g}}$$

Precession angular rate at latitude $\varphi$:

$$\Omega_{\mathrm{pre}} = \Omega_\oplus \sin|\varphi|,\qquad \Omega_\oplus = \dfrac{2\pi}{86164.1\,\mathrm{s}} \approx 7.292\times10^{-5}\,\mathrm{rad/s}$$

Full-rotation time and accumulated angle after time $t$:

$$T_{\mathrm{pre}} = \dfrac{2\pi}{\Omega_{\mathrm{pre}}},\qquad \Delta\theta = \Omega_{\mathrm{pre}}\,t$$

$L$ is the pendulum length [m] and $g$ is the local gravity [m/s²]. The precession depends only on $\sin\varphi$ — not on $L$ or mass — so the rotation is clockwise in the northern hemisphere, counterclockwise in the southern, and zero on the equator.

What is the Foucault Pendulum Simulator

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I saw a Foucault pendulum at a science museum once and the swing direction kept drifting. Is the pendulum itself rotating?

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Great question. The pendulum keeps swinging in the same direction in absolute space. What is moving is the floor — the Earth turns once a day, so for someone standing on the floor the swing plane appears to drift the opposite way. In 1851 Foucault hung a 67 m, 28 kg pendulum at the Pantheon in Paris and gave the first laboratory-scale visual proof that the Earth rotates.

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When I set latitude to 0 (equator) the rotation goes away. At 90 (the pole) it makes one full turn per day.

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That is exactly the law. The precession rate is $\Omega_\oplus \sin\varphi$, proportional to the sine of latitude. At the pole sin90 = 1, one full rotation per sidereal day (23.93 h). In Tokyo near 35 degrees, sin35 ~ 0.574, so about 41.7 h per rotation. At the equator sin0 = 0 and there is no rotation. A pendulum can literally tell you "where on Earth you are" — pretty striking, right?

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If I make the pendulum longer, does the rotation speed up?

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No. The precession depends only on latitude — not on length L, mass, or amplitude. L only sets the swing period $T = 2\pi\sqrt{L/g}$, so longer pendulums simply swing more slowly. The reason real Foucault pendulums use 60–70 m wires is to delay air-drag damping with a heavier bob and longer wire, so the swing keeps going for many hours and the slow precession becomes visible.

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I keep hearing about the "Coriolis force." Is it related?

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Same physics, different viewpoint. From the rotating Earth frame the Coriolis force $F = -2m\,\boldsymbol{\Omega}\times\boldsymbol{v}$ acts perpendicular to the velocity. In the northern hemisphere it always pushes to the right of the swing, gradually turning the plane clockwise; in the southern hemisphere it pushes to the left, counterclockwise. Same mechanism that organizes hurricane spin and ballistic deflection.

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"Precession" — I have heard the same word for spinning tops?

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Good catch. In a broad sense "precession" is any slow rotation of an axis. The wobble of a top under gravity, the 26000-year precession of the equinoxes from Earth's axial tilt, and the rotation of the Foucault swing plane all share the same mathematical structure. Late in a classical-mechanics course you'll watch all of these collapse into one beautiful framework.

FAQ

The Foucault swing plane stays fixed in an inertial frame (the frame of the distant stars), so the rotation rate of the floor must be measured in that same frame — that is the sidereal day, 86164.1 s. The familiar 86400 s day (the solar day) is the time for the Earth to face the Sun the same way again, and it includes about one extra degree of orbital motion. So at the pole the Foucault pendulum makes one full revolution in 23.934 h — sidereal, not solar.

The rotation reverses to counterclockwise. The magnitude is still |sin φ|, but the sign of the Coriolis force flips. In Sydney (about −34°) the swing plane rotates counterclockwise with about a 43-hour period, while at the equator sin0 = 0 produces no rotation. Slide the latitude in the simulator to a negative value: the orange arc visibly reverses. Real Foucault pendulums in Buenos Aires and Melbourne rotate in the opposite sense from those in northern museums.

In Japan: the National Museum of Nature and Science in Tokyo, the Kyoto University museum, and Nagoya City Science Museum. In the world: the Conservatoire des Arts et Métiers (CNAM) and the Pantheon in Paris (replica) hold the original Foucault pendulums. Most science museums use a magnetic drive to compensate damping and arrange a circle of marker pegs so visitors can watch them topple in sequence — a striking visual log of the precession at that latitude.

The simulator uses (1) a small-amplitude simple-pendulum approximation, (2) the linear Coriolis theory that decouples swinging from precession, and (3) zero air drag and friction. Real pendulums show added effects: large amplitudes generate elliptical orbits that mimic spurious precession (the Majorana effect), air drag damps the swing, and temperature changes the wire length. To measure Omega*sin(phi) precisely, amplitude, temperature, and initial velocity must be tightly controlled. Even so, this formula matches real pendulums within about 1 percent.

Real-world applications

A direct demonstration of Earth rotation: Foucault's 1851 experiment at the Pantheon was the first laboratory-scale visual proof of Earth rotation, an effect previously confirmed only through astronomical observations. Modern science museums still use the Foucault pendulum as a flagship exhibit because the latitude dependence of precession lets visitors literally see that the Earth is a rotating sphere. Pedagogically it teaches the deep physics idea that "we infer motion from things that do not move."

Inertial navigation systems (INS): Mechanical gyrocompasses on submarines, aircraft, and missiles exploit the same "axis stays fixed in an inertial frame" property as the Foucault pendulum. A spinning top senses Earth rotation and aligns to a meridian precisely because Omega*sin(phi) creates a torque about the local horizontal. Where GPS is not available — deep underwater, in space, in electronic-warfare environments — modern descendants such as ring-laser and fiber-optic gyros remain the workhorses.

Coriolis analysis in meteorology and oceanography: Hurricanes spin counterclockwise in the northern hemisphere (surface convergence) and clockwise in the southern hemisphere because the Coriolis parameter Omega*sin(phi) flips sign across the equator — the same physics as the pendulum. Operational numerical weather and ocean general-circulation models embed this "Foucault term" directly in the equations of motion. Sliding the latitude control here gives direct intuition for why hurricanes form mostly outside the equator.

Seismology and Earth-interior structure: The Earth's free oscillations — long-period seismic modes — are slightly split in frequency because rotation imposes a Foucault-like precession on each mode. Observed splitting widths constrain the differential rotation of the inner core and the structure of the deep interior. Global seismic networks built up since the 1960s have used this Foucault-style effect to refine our picture of the Earth's interior.

Common misconceptions and caveats

The most common misconception is to believe that the pendulum itself rotates. In reality the swing plane stays fixed in an inertial frame and our feet (the Earth) rotate beneath it; the apparent rotation is purely a frame effect. The top-down canvas in this simulator overlays the inertial-frame swing axis (blue) and the t = 0 ground reference (dashed), but the picture is drawn from the ground frame, so they appear to drift relative to each other — keep this dual view in mind.

The second is to assume that pendulum length or mass changes the rotation rate. Move the L or g sliders and the precession value (°/h) does not budge — only the swing period T_osc changes. The 60–70 m lengths of real Foucault pendulums are not chosen for faster rotation; they reduce air-drag losses (a gentler swing dissipates less energy), so the swing keeps going long enough for the slow precession to become visually obvious.

Finally it is wrong to dismiss the equatorial case as uninteresting. Yes, the precession rate is zero because it is proportional to sin(latitude), but near the equator the horizontal component of Omega is maximum and produces a different perturbation: the Majorana effect, in which the swing spontaneously turns elliptical. So the correct distinction is "vertical-axis precession is zero, but in-plane ellipticity is maximum," and a perfectly straight-line swing is impossible to maintain in practice.

Foucault Pendulum Simulator — Earth Rotation Precession

What is the Foucault Pendulum Simulator

FAQ

Real-world applications

Common misconceptions and caveats

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