Foucault Pendulum (Earth's Rotation) — 3D Lab Simulation
Visual proof of Earth's rotation via Coriolis precession
Theoretical Foundation & Historical Context
At the Panthéon in Paris, Foucault suspended a 28kg brass bob from a 67m steel wire. The plane of swing appeared to slowly rotate relative to the floor, visually proving that Earth itself rotates beneath the pendulum.
Law: Coriolis Force & Non-Inertial Reference Frames | Scientist: Léon Foucault | Year: 1851
Key Learning Objectives & Competencies
- ✓ Prove the diurnal rotation of Earth through simple mechanical oscillation inside a terrestrial laboratory.
- ✓ Calculate the latitude-dependent precession rate ω = Ω·sin(φ) and total daily rotation angle.
- ✓ Analyze the apparent Coriolis fictitious force operating in a rotating reference frame.
Mathematical Formulations & Differential Equations
ω_prec = Ω_Earth · sin(φ) = 15°/h · sin(φ)
Directly proportional to sine of observer latitude
T_prec = 24h / |sin(φ)|
Poles = 24h, Paris = 31.8h, Equator = infinite
T₀ = 2π · √(L / g)
For L = 67m, period T₀ is approx 16.5s
Physical Parameters & Units of Measurement
| Symbol | Physical Quantity | SI Unit | Default |
|---|---|---|---|
| φ | Observer geographic latitude (°) | — | — |
| L | Pendulum suspension wire length (m) | — | — |
| Ω_Earth | Earth rotation angular speed (15°/h) | — | — |
Real-World & Industrial Engineering Applications
Inertial navigation systems (INS) for aircraft, nuclear submarines, and spacecraft trajectory guidance.
Atmospheric vortex dynamics, cyclone modeling, and ocean current gyre prediction.
Gyrocompass engineering providing true geographic North without magnetic compass declination.
3-Step Controlled Experimental Protocol
- Step 1: Baseline Setup — Initialize the simulation in standard conditions and observe undisturbed system dynamics.
- Step 2: Parameter Variation — Gradually modulate physical parameters using the control panel sliders.
- Step 3: Verification & Phase Space Analysis — Compare phase space trajectories and conservation metrics against analytical ODE solutions.
Frequently Asked Questions (FAQ)
Why does the plane of oscillation appear to rotate rather than the pendulum itself?
In an inertial frame of reference, the pendulum's swinging plane remains fixed in space while the Earth rotates underneath it. To observers standing on the rotating Earth, the plane appears to slowly precess.
What is the precession period of a Foucault pendulum at the North Pole vs the Equator?
At the North Pole (φ = 90°), the pendulum completes a full 360° rotation in exactly 24 sidereal hours (23h 56m). At the Equator (φ = 0°), sin(0) = 0, so the plane of oscillation does not precess at all.
Why did Foucault use a 67-meter cable and a 28-kilogram brass bob in 1851?
A long cable gives a slow oscillation period (T ≈ 16.5 seconds) minimizing air turbulence, while a heavy bob provides immense momentum to overcome air resistance and maintain oscillation for hours without stopping.
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