Foucault Pendulum

Foucault Pendulum. A Foucault pendulum is a long, lightly damped oscillatory system whose bob is free to swing in any vertical plane while the apparatus is fixed to the rotating Earth. The restoring force is gravity, and the motion can be treated in the small‑angle approximation by a Lagrangian that separates the planar motion from Earth’s rotation. The pendulum’s angular momentum about its vertical axis is conserved in the local inertial frame, but the Earth’s rotation introduces a Coriolis torque that causes the swing plane to precess. The precession rate is ω = Ω sin φ, where Ω ≈ 7.292 × 10⁻⁵ rad s⁻¹ is the Earth’s angular velocity and φ is the latitude of the pendulum’s location. At the North or South Pole the plane completes a full revolution in 24 h, while at the equator no precession occurs. The effect is most apparent in large pendulums (arm lengths > 10 m) because the period of swing is longer, giving the Earth more time to rotate beneath the bob between successive swings, and because the ratio of restoring torque to inertial resistance is favorable for observing the subtle precessional motion.

Theoretical Context

The first practical demonstration was made by Léon Foucault in 1851 at the Paris Observatory, using a 67‑m pendulum that visibly rotated its swing plane in a matter of hours. Subsequent installations around the globe, including the Smithsonian Institution’s 14‑m Harvard pendulum and the 12‑m Chicago–University of Illinois model, were designed to mitigate air resistance, ground vibrations, and buoyancy effects to maintain a nearly frictionless environment. These experiments not only confirmed the rotation of the Earth but also served as sensitive tests of the conservation of angular momentum in a non‑inertial reference frame. Modern incarnations use air‑tight, vacuum‑sealed cases, magnetic suspensions, or optical interferometry to further reduce energy loss, extending the observable precession period and enabling educational demonstrations of fundamental rotational dynamics.