M059 · Earth rotation / two timescales

Foucault Pendulum & Earth Rotation

Why does a Foucault pendulum's length change its swing period but not how fast its swing plane turns? Because the two have different causes. Length sets how quickly each swing repeats. The swing plane turns because Earth rotates beneath it, and only the part of that rotation about the local vertical counts, which depends on latitude. At the poles the plane turns once a day; at the equator it does not turn. Changing the length leaves that rate alone.

Turn the three-dimensional view around the support, cable and bob, and magnify the motion if you like. Move from the equator to either pole and follow hours of precession alongside single swings. Compare the local coordinate views, measure the precession rate by sampling the bob once per swing, and see how much energy damping removes.

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 $\text{Pending}$

Watch the motion and compare the cases

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 set the starting displacement and direction, or use the sliders, which also work from the keyboard. Step forward one eighth of a swing at a time to follow the fast motion; use the horizon scrubber to follow the slow precession. Trails show only the last two fully resolved swings. The teal phase probes sample the actual position after each whole fast cycle; they measure the phase drift, not a fitted ellipse axis. The modal carrier phase is unwrapped, while a swing axis has no direction and repeats after a half turn. Nearly circular motion has no well-defined swing plane. Probes smaller than one millionth of the initial displacement are discarded. Sparse energy points and guide lines cover the full record; playback compresses time, so fast swings may look aliased.

The model is a small-amplitude pendulum that behaves the same in every horizontal direction, with constant effective gravity and linear damping relative to the ground. The apparatus magnifies horizontal motion eight times by default; turn magnification off to see true proportions. The cable height is worked out from geometry and adds no vertical motion. Only the vertical component of Earth’s rotation enters the Coriolis term. Higher-order centrifugal corrections that vary with position, vertical coupling, finite-amplitude Airy precession, anisotropy and drive are left out. The analytic solution is exact for this linear model, not for the full spherical pendulum. The sidereal day is a fixed, rounded value. The second coordinate view removes the precession in a local frame; it is not a global inertial frame.

Physics tutorial

Read a day of rotation from a few seconds of swinging

BackgroundThe pendulum swings in seconds, but its swing plane turns over hours. Over the small patch of floor the bob covers, the Coriolis coupling in the model comes from the vertical component of Earth’s rotation. The constant local centrifugal effect is already part of effective gravity.

Why it mattersA long pendulum lets you measure Earth’s rotation in one place. It also shows how to pull a slow phase out of a fast oscillation, and why damping or a nearly circular release can hide the swing axis you are trying to watch.

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 how fast the pendulum swings. Latitude sets how fast the swing plane turns, and in which direction. Checking the bob’s position once per full swing reveals the slow turn without joining up widely spaced samples of the fast motion.

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 Lab uses a fixed sidereal day as a teaching constant; it is not the same as a solar day. At positive latitudes the swing plane turns clockwise when seen from above.

Horizontal equations of motion

\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 points along the positive horizontal axis and north points up the plot. Damping acts on the velocity relative to the ground. Solving these linear equations exactly does not bring back the spherical and finite-amplitude effects they leave out.

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}

The full initial position and velocity fix the complex amplitudes. Each decaying mode is computed separately so that long damped records do not overflow.

Turn of the swing plane and its full-turn period

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

The plot keeps counting whole turns. A swing axis has no head or tail, so it looks the same again after half this period. At the equator the model gives 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 energies per unit mass in the linear horizontal model. The Coriolis acceleration does no work; linear damping turns mechanical energy into dissipated energy.

Sample the position once per swing

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

The bob’s position angle is read after each whole fast cycle and unwrapped without using the precession formula. Without damping, these angles give back the formula’s rate. Damping can add a small phase bias between the modes. The probe tracks position angles; it does not fit the axis of an ellipse.

Common difficulties

Longer means faster precession

Typical misconceptionA longer pendulum should precess at a different rate at the same place.

Better mental modelLength changes the swing period. In this model the precession rate depends on latitude and Earth’s rotation, not on length.

The second view is globally inertial

Typical misconceptionTaking out the visible precession gives the view of an inertial observer at Earth’s centre.

Better mental modelThe second view only rotates the local horizontal axes. Away from the poles, the local vertical itself changes direction in space.

Any moving point defines a swing plane

Typical misconceptionThe direction of the bob right now is the precession angle, even when it moves in a near circle.

Better mental modelThe fast swing and the slow turn of the swing plane are different things. A nearly circular orbit has no clear long axis, and the Lab flags when the axis cannot be resolved.

Numerical exactness removes physical approximations

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

Better mental modelThe solution is exact only for the isotropic linear model. Over long observations, effects it leaves out can build up: finite-amplitude Airy precession, anisotropy, drive and higher-order Earth terms.

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 trail covers only the last two swings, sampled finely enough to resolve them. The Lab does not draw a day-long bob path from sparse samples.
02
Compare hemispheres
Review the equator, both hemispheres and the North Pole. Compare the measured phase-probe rate with the analytic carrier rate.
What to observe: The swing plane does not turn at the equator, turns the other way across it, and makes one 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, but the precession rate at this latitude stays the same.
04
Watch the swing plane become hard to see
Compare an elliptical release, a nearly circular release and the damped day-long record.
What to observe: In the near-circular case the axis is poorly defined. Damping shrinks the real swing, measured in metres, while mechanical energy plus dissipated energy stays constant in the model.