---
title: "Foucault Pendulum & Earth Rotation Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/foucault-pendulum/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/foucault-pendulum/"
last_modified: "2026-10-22"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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

**Background** The 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 matters** A 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 misconception** A longer pendulum should precess at a different rate at the same place.

**Better mental model** Length 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 misconception** Taking out the visible precession gives the view of an inertial observer at Earth’s centre.

**Better mental model** The 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 misconception** The direction of the bob right now is the precession angle, even when it moves in a near circle.

**Better mental model** The 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 misconception** An exact solution proves the real pendulum follows this curve for days.

**Better mental model** The 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

1. 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.

2. 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.

3. 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.

4. 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.

Physical conclusion

Resolve the fast swing first, then measure the slow turn, and keep track of what the model leaves out. The turning plane you see is evidence within this model; it does not replace modelling every correction a real pendulum needs.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)