Foucault Pendulum Simulator

1. Real-world phenomenon

In 1851 Léon Foucault suspended a 67-meter wire from the dome of the Panthéon and set a heavy bob swinging. Hours later onlookers noticed that the plane of oscillation had rotated. The pendulum itself was not twisting; instead, Earth had turned beneath it. This simple yet profound demonstration provided palpable evidence of our planet's rotation, capturing the public imagination in an era when heliocentrism was still controversial in some circles. Modern science museums continue the tradition with smaller replicas that precess slowly through the day. A typical calculator might tell you how many hours it takes for the swing plane to complete a full turn at a given latitude, but it offers no sense of the graceful motion. The simulator above bridges that gap: a canvas shows a top-down view of the bob tracing ellipses that rotate over time, revealing Earth's influence at a glance. Energy bars track the trade between kinetic and potential energy, while a hidden text summary narrates the motion for screen readers. By adjusting latitude, pendulum length, and a time-scaling factor that accelerates the otherwise imperceptible precession, you can explore how the Coriolis effect manifests in this iconic experiment.

2. Variables and assumptions

The model assumes a point mass bob of negligible size attached to a massless, rigid wire of length $L$. The bob's horizontal displacement $(x,y)$ from the suspension point remains small compared with $L$, so the small-angle approximation holds. Gravity $g=9.81$ m/s² acts downward, producing a natural oscillation frequency $\mathrm{\omega \_0}=\surd (g/L)$. Earth's rotation rate is $\mathrm{\Omega \_E}=7.292\times 10^-5$ rad/s; only the component perpendicular to the local surface, $\mathrm{\Omega \_E\; sin\phi }$, influences the pendulum. A dimensionless time-scaling factor multiplies this rotation rate to make the precession visible within minutes rather than hours; setting the scale to 1 recovers real time. Damping and driving forces are neglected, so the motion is conservative. All inputs are in SI units, and fields are validated for finiteness and non-negative length and time step. The amplitude is clamped to keep the bob within view, reinforcing that the small-angle assumption limits $\surd (x^2+y^2)\; \ll \; L$.

3. Governing equations

In a rotating reference frame, the Coriolis force couples the horizontal motions. Let $f=2\; \Omega \_E\; sin\phi$ denote the Coriolis parameter. The equations of motion are

$\frac{\mathrm{d^2x}}{\mathrm{dt^2}}=f\frac{\mathrm{dy}}{\mathrm{dt}}-{\mathrm{\omega \_0}}^{2}x$, $\frac{\mathrm{d^2y}}{\mathrm{dt^2}}=-f\frac{\mathrm{dx}}{\mathrm{dt}}-{\mathrm{\omega \_0}}^{2}y$

Velocity components satisfy $\frac{\mathrm{dx}}{\mathrm{dt}}=v\_x$ and $\frac{\mathrm{dy}}{\mathrm{dt}}=v\_y$. Kinetic energy for unit mass is $\mathrm{KE}=0.5(v\_x^2+v\_y^2)$, while potential energy arises from vertical displacement: $\mathrm{PE}=0.5(\omega \_0^2)(x^2+y^2)$. In the absence of damping, total energy $\mathrm{TE}=\mathrm{KE}+\mathrm{PE}$ should remain constant; deviations highlight numerical error. The theoretical precession rate of the swing plane is simply $\Omega \_E\; sin\phi$, yielding a period $\frac{\mathrm{2\pi }}{\Omega \_E\; sin\phi }$.

4. Numerical scheme

The coupled equations form a linear system with periodic coefficients. Although analytic solutions exist, the simulator integrates them numerically to expose algorithmic considerations. A classical fourth-order Runge–Kutta scheme updates the state vector $(x,y,v\_x,v\_y)$ every step $\mathrm{\Delta t}$. Four intermediate slopes sample the Coriolis-modified accelerations, and their weighted average advances the solution. RK4's local error scales with ${\mathrm{\Delta t}}^5$, making it both accurate and straightforward. The time step is clamped between 0.0005 and 0.1 s; smaller values improve fidelity but require more computations, while larger values can destabilize the integration, especially when the Coriolis term is magnified by large time scales. An energy deviation readout compares $\mathrm{TE}$ to its initial value, revealing drift that grows with $\mathrm{\Delta t}$ and providing a concrete lesson in numerical stability. The time-scaling factor multiplies the Coriolis parameter but does not alter $\mathrm{\Delta t}$, so the integrator still operates in SI seconds.

5. Worked example

Imagine a 10 m pendulum located at latitude $45°$ with an initial eastward displacement of 0.5 m. Set the time-scaling factor to 1000 and the step size to 0.02 s. Press Play, and the bob sweeps out an ellipse that slowly rotates clockwise. The summary reports the simulated time and the current azimuth of the oscillation. After 60 s of simulated time (equivalent to about one minute/1000 ≈ 0.06 real seconds of Earth rotation), the plane has rotated roughly $60\; \times \; \Omega \_E\; sin45°\; \times \; 1000\; \approx \; 3.1$ radians, or half a turn, making the effect dramatic. Reset and set the scale to 1 to observe the true pace: the ellipse barely twists over the same interval, illustrating why full-size exhibits require hours to show a noticeable change. Exporting the CSV yields columns of time, x, y, velocity components, and energies that can be plotted to reveal Lissajous figures and confirm energy conservation.

6. Comparison table

The table contrasts the baseline example with two variants: one at the equator and one at higher latitude. The precession period is computed from $\frac{\mathrm{2\pi }}{\Omega \_E\; sin\phi }$.

Latitude (deg)	Time Scale	Precession Period (h)	Simulated 60 s Rotation (deg)
45	1000	23.93	180
0	1000	∞	0
60	1000	19.95	207

At the equator the precession period diverges and the simulated rotation vanishes, matching theory. Moving to 60° latitude shortens the period and yields a faster twist over the same 60-second span.

7. How to read the animation

The canvas shows a bird's-eye view with the suspension point at center. A blue trail records the bob's path; an orange square marks its current location. The scale adjusts to keep the entire trajectory visible, and the axes are omitted to emphasize relative motion. The orange striped bar beneath the canvas indicates kinetic energy, while the blue bar tracks potential energy; together their lengths equal the initial total energy. The caption narrates time, position, and energy drift in words, mirrored by the hidden text summary for accessibility. With keyboard focus on the canvas, the space bar toggles play and pause. Because colors are paired with textures and textual labels, viewers with color vision deficiencies can still interpret the energy bars.

8. Limitations

The model neglects air resistance and the Earth's non-spherical shape. Real pendulums suffer damping that gradually shrinks the amplitude, often requiring electromagnetic drivers to sustain motion; our energy bars remain constant because we omit these losses. Large amplitudes violate the small-angle approximation, leading to period elongation and altered precession; the code clamps amplitude to keep the approximation plausible. The time-scaling factor is a pedagogical convenience but divorces the animation from physical time; values much larger than 1000 may introduce numerical artifacts if the effective Coriolis term becomes comparable to the natural frequency. Finally, the simulation is confined to horizontal motion and does not render the vertical swing, so it cannot display conical pendulum behavior.

9. Suggested extensions

Enhancements could include a damping coefficient to demonstrate how friction gradually erodes energy, or an automated driver that keeps the amplitude constant as in museum exhibits. Incorporating Earth's elliptical shape or latitude-dependent gravitational acceleration would refine the period calculation. A three-dimensional rendering showing the bob's vertical motion could complement the top-down view. Another extension would compare the numeric precession rate to the analytic $\Omega \_E\; sin\phi$ in real time, highlighting any divergence caused by large amplitudes or numerical error.

10. References and related tools

Detailed treatments of Foucault pendulums appear in textbooks such as Marion and Thornton's Classical Dynamics of Particles and Systems and M. L. Barton's Introduction to Geophysical Fluid Dynamics. For more on rotating frames and the Coriolis effect, consult any fluid dynamics text. Explore related calculators here: determine the period of small oscillations with the Simple Pendulum Period Calculator, compare rotational effects with the Sagnac Effect Simulator, or analyze damping in the Mass-Spring Oscillation Simulator.