M059 · Earth rotation / two timescales

Foucault Pendulum & Earth Rotation

Orbit the pendulum support, cable and bob in three dimensions, with optional motion magnification. Move from the equator to either pole and follow hours of precession alongside individual swings. Compare local coordinate views, extract a rate from phase-locked trajectory probes, and account for dissipated energy.

Reviewed time $\text{Pending}$

Latitude $\text{Pending}$

Vertical Earth rotation component $\text{Pending}$

Fast oscillation period $\text{Pending}$

Analytic local carrier rate $\text{Pending}$

Directed full-turn period $\text{Pending}$

Unwrapped carrier turn $\text{Pending}$

Local modal major semiaxis $\text{Pending}$

Minor-to-major semiaxis ratio $\text{Pending}$

Ground-relative bob speed $\text{Pending}$

Mechanical energy fraction $\text{Pending}$

Dissipated energy fraction $\text{Pending}$

Normalized energy balance defect $\text{Pending}$

Coriolis acceleration magnitude $\text{Pending}$

Omitted planar period correction estimate $\text{Pending}$

Swing-axis observability $\text{Pending}$

Measured phase-probe rate $\text{Pending}$

Reviewed valid phase probes $\text{Pending}$

OBSERVE / COMPARE / VERIFY

Can a pendulum reveal Earth’s rotation?

Latitude

\varphi

48.86\,\mathrm{{}^\circ}

Pendulum length

L

67.00\,\mathrm{m}

Initial displacement / length

A/L

0.03\,

Release direction from east

\alpha_0

30.00\,\mathrm{{}^\circ}

Transverse launch velocity ratio

v_\perp/(A\omega_0)

0.00\,

Amplitude decay time; zero means none

\tau_A

0.00\,\mathrm{h}

Observation horizon

T

12.00\,\mathrm{h}

Reviewed fraction of the horizon

t/T

0.00\,\%

Drag the orange release ring to change displacement and direction, or use the keyboard-accessible sliders. Advance one eighth of a swing to resolve the fast motion; the horizon scrubber resolves slow precession. Trails show only the last two resolved swings. Teal phase probes sample actual positions after whole fast cycles; they measure phase drift, not an ellipse-axis fit. The modal carrier is unwrapped; an unoriented swing axis repeats after a half turn. Near-circular motion has no well-resolved plane. Probes below one millionth of the initial displacement are discarded. Sparse energy points and guide lines show the full record; playback compresses time and may visually alias fast swings.

Small-amplitude isotropic horizontal pendulum with constant effective gravity and linear damping relative to the ground. The apparatus defaults to eightfold horizontal magnification; turn it off for true proportions. Cable height is reconstructed geometrically and does not add vertical dynamics. Only the vertical component of Earth’s rotation enters the retained Coriolis term; higher-order spatial centrifugal corrections, vertical coupling, finite-amplitude Airy precession, anisotropy and drive are omitted. The analytic solution is exact for this linear model, not the full spherical pendulum. A fixed rounded sidereal day is used. The second coordinate view is a de-precessed local reference, not a global inertial frame.

Physics tutorial

Read a day of rotation from a few seconds of swinging

BackgroundThe pendulum swings in seconds while its local carrier turns over hours. In a small horizontal patch, the vertical projection of Earth’s rotation produces the retained Coriolis coupling. Effective gravity already includes the constant local centrifugal contribution.

Why it mattersA long pendulum turns Earth’s rotation into a local measurement. The experiment also teaches how to separate a fast oscillation from a slow phase, and why damping and a nearly circular release can hide the observable swing axis.

Start with the essentials

Focus question: Does changing the length change the precession rate, and how can the rate be recovered from actual bob positions?
One-sentence intuition: Length sets the fast oscillation period. Latitude sets the signed local carrier rate. Phase-locked position probes reveal the slow turn without connecting widely spaced fast-motion samples.

Core mathematical model

Two time scales

\omega_0^2=g_{\rm eff}/L,\qquad\nu=\Omega_\oplus\sin\varphi,\qquad\Omega_\oplus=\frac{2\pi}{23.9345\times3600\,\mathrm s}

The fixed sidereal teaching period differs from a solar day. Positive latitude means a clockwise carrier turn when viewed from above.

Retained horizontal dynamics

\ddot x+2\gamma\dot x+\omega_0^2x-2\nu\dot y=0,\qquad\ddot y+2\gamma\dot y+\omega_0^2y+2\nu\dot x=0

East is the positive horizontal axis, north the positive vertical plot axis. Damping acts on ground-relative velocity; omitted spherical and finite-amplitude effects are not recovered by an exact linear solution.

Exact damped modal solution

z=c_+e^{\lambda_+t}+c_-e^{\lambda_-t},\qquad\lambda_\pm=-(\gamma+i\nu)\pm\sqrt{(\gamma+i\nu)^2-\omega_0^2}

Complex amplitudes are fixed by the full initial position and velocity. Separate decaying modes avoid overflow in long damped records.

Local carrier and directed period

\Delta\psi=-\nu t,\qquad T_{\rm full}=\frac{23.9345\,\mathrm h}{|\sin\varphi|},\qquad\varphi\ne0

The plotted carrier retains full turns. An unoriented swing axis repeats after half this directed period. At the equator the retained model has no Foucault precession.

Mechanical energy and heat

e=\tfrac12(|\dot z|^2+\omega_0^2|z|^2),\qquad q=2\gamma\int_0^t|\dot z|^2\,dt,\qquad e+q=e_0

These are energy per mass in the linear horizontal model. Coriolis acceleration does no work; linear damping converts mechanical energy into dissipated energy.

Resolve swings and probe their phase

T_{\rm fast}=\frac{2\pi}{\operatorname{Im}\sqrt{(\gamma+i\nu)^2-\omega_0^2}},\qquad t_n=nT_{\rm fast}

Position angles at whole fast cycles are unwrapped independently of the carrier formula. With no damping they recover its rate. Damping can introduce a small modal phase bias; this probe is not an ellipse-axis fit.

Common difficulties

Longer means faster precession

Typical misconceptionA longer pendulum should precess at a different geographic rate.

Better mental modelLength changes the fast swing period, while the retained carrier rate depends on latitude and Earth’s rotation.

The second view is globally inertial

Typical misconceptionRemoving the visible precession constructs an Earth-centred inertial observer.

Better mental modelThe second view applies only a local horizontal coordinate rotation. Away from the poles, the local vertical itself changes direction in space.

Any moving point defines a swing plane

Typical misconceptionThe bob’s current azimuth is the precession angle, even for a nearly circular orbit.

Better mental modelFast oscillation and slow carrier phase are different. Nearly circular motion has no reliably resolved major axis; the observability readout identifies this limit.

Numerical exactness removes physical approximations

Typical misconceptionAn exact solution proves the real pendulum follows this curve for days.

Better mental modelIt is exact only for the isotropic linear model. Finite-amplitude Airy precession, anisotropy, drive and higher-order Earth terms can accumulate over long observations.

Run the experiment

01
Resolve one swing
Select the short pendulum. Advance in eighth-swing steps and inspect the ground and local reference views.
What to observe: The recent trail is resolved within two fast periods; no sparsely sampled day-long bob curve is drawn.
02
Compare hemispheres
Review the equator, both hemispheres and the North Pole. Compare the measured position-probe rate with the carrier.
What to observe: The turn vanishes at the equator, changes sign across it and reaches one directed full turn per sidereal day at a pole.
03
Change length without changing latitude
Use the length slider, then drag the orange release ring to change direction and amplitude.
What to observe: The swing period and displacement change, while the geographic carrier rate remains the same.
04
Lose the observable plane
Compare an elliptical release, a nearly circular release and the damped day-long record.
What to observe: The near-circle has a weakly defined axis. Damping reduces the physical metre-scale signal, while energy plus dissipation remains constant in the model.