The Foucault pendulum ("foo-KOH"), or Foucault's pendulum, named after the French physicist Léon Foucault, was conceived as an experiment to demonstrate the rotation of the Earth.

The experiment

The experimental apparatus consists of a tall pendulum free to oscillate in any vertical plane. The direction along which the pendulum swings rotates with time because of Earth's daily rotation. The first public exhibition of a Foucault pendulum took place in February 1851 in the Meridian Room of the Paris Observatory. A few weeks later, Foucault made his most famous pendulum when he suspended a 28-kg bob with a 67-metre wire from the dome of the Panthéon in Paris. The plane of the pendulum's swing rotated clockwise 11° per hour, making a full circle in 32.7 hours.

In 1851 it was well known that Earth rotated: observational evidence included Earth's measured polar flattening and equatorial bulge. However, Foucault's pendulum was the first dynamic proof of the rotation in an easy-to-see experiment, and it created a sensation in both the learned and everyday worlds.

At either the North Pole or South Pole, the plane of oscillation of a pendulum remains fixed with respect to the fixed stars while Earth rotates underneath it, taking one sidereal day to complete a rotation. So relative to Earth, the plane of oscillation of a pendulum at the North or South Pole undergoes a full clockwise or counterclockwise rotation during one day, respectively. When a Foucault pendulum is suspended on the equator, the plane of oscillation remains fixed relative to Earth. At other latitudes, the plane of oscillation precesses relative to Earth, but slower than at the pole; the angular speed, $alpha$ (measured in clockwise degrees per sidereal day), is proportional to the sine of the latitude, $phi$:

$alpha=360,sin(phi).$

Here, latitudes north and south of the equator are defined as positive and negative, respectively. For example, a Foucault pendulum at 30° south latitude, viewed from above by an earthbound observer, rotates counterclockwise 180° in one day.

In order to demonstrate the rotation of the Earth without the philosophical complication of the latitudinal dependence, Foucault named the gyroscope in 1852. The gyroscope's spinning rotor tracks the stars directly. Its axis of rotation is observed to return to its original orientation with respect to the earth after one day whatever the latitude, unaffected by the sine factor.

A Foucault pendulum requires care to set up because imprecise construction can cause additional veering which masks the terrestrial effect. The initial launch of the pendulum is critical; the traditional way to do this is to use a flame to burn through a thread which temporarily holds the bob in its starting position, thus avoiding unwanted sideways motion. Air resistance damps the oscillation, so Foucault pendulums in museums often incorporate an electromagnetic or other drive to keep the bob swinging; others are restarted regularly. In the latter case, a launching ceremony may be performed as an added show.

The dynamics of the Foucault pendulum

From the perspective of an inertial frame outside of Earth, the suspension point of the pendulum traces out a circular path during one sidereal day. No forces act to make the plane of oscillation of the pendulum rotate - the plane contains the plumb line, so the force acting on the pendulum is parallel to the plane of oscillation at all times. But the plane satisfies the constraint that it contains the plumb line. Thus the plane of oscillation undergoes parallel transport. The difference between initial and final orientations is $alpha=-2pi,sin(phi)$ as given by the Gauss-Bonnet theorem. $alpha$ is also called the holonomy or geometric phase of the pendulum. Thus, when analyzing earthbound motions, the Earth frame is not an inertial frame, but rather rotates about the local vertical at an effective rate of $2pi,sin(phi)$ radians per day, which is the magnitude of the projection of the angular velocity of Earth onto the normal direction to Earth.

From the perspective of an Earth-bound coordinate system with its $x$-axis pointing east and its $y$-axis pointing north, the precession of the pendulum is explained by the Coriolis force. Consider a planar pendulum with natural frequency $omega$ in the small angle approximation. There are two forces acting on the pendulum bob: the restoring force provided by gravity and the wire, and the Coriolis force. The Coriolis force at latitude $phi$ is horizontal in the small angle approximation and is given by

$$

begin{align} F_{c,x} &= 2 m Omega dfrac{dy}{dt} sin(phi) F_{c,y} &= - 2 m Omega dfrac{dx}{dt} sin(phi) end{align} where $Omega$ is the rotational frequency of Earth, $F\_\{c,x\}$ is the component of the Coriolis force in the x-direction and $F\_\{c,y\}$ is the component of the Coriolis force in the y-direction.

The restoring force, in the small angle approximation, is given by

$$

begin{align} F_{g,x} &= - m omega^2 x F_{g,y} &= - m omega^2 y. end{align}

Using Newton's laws of motion this leads to the system of equations

$$

begin{align} dfrac{d^2x}{dt^2} &= -omega^2 x + 2 Omega dfrac{dy}{dt} sin(phi) dfrac{d^2y}{dt^2} &= -omega^2 y - 2 Omega dfrac{dx}{dt} sin(phi). end{align}

Switching to complex coordinates $z=x+iy$ the equations read

$$

dfrac{d^2z}{dt^2} + 2iOmega dfrac{dz}{dt} sin(phi)+omega^2 z=0.

To first order in $Omega/omega$ this equation has the solution

$$

z=e^{-iOmega sin(phi)t}left(c_1 e^{iomega t}+c_2 e^{-iomega t}right).

If we measure time in days, then $Omega=2pi$ and we see that the pendulum rotates by an angle of $-2pi,sin(phi)$ during one day.

Related physical systems

There are many physical systems that precess in a similar manner to a Foucault pendulum. In 1851, Charles Wheatstone described an apparatus that consists of a vibrating spring that is mounted on top of a disk so that it makes a fixed angle $phi$ with the disk. The spring is struck so that it oscillates in a plane. When the disk is turned, the plane of oscillation changes just like the one of a Foucault pendulum at latitude $phi$.

Similarly, consider a non-spinning perfectly balanced bicycle wheel mounted on a disk so that its axis of rotation makes an angle $phi$ with the disk. When the disk undergoes a full clockwise revolution, the bicycle wheel will not return to its original position, but will have undergone a net rotation of $2pi,\; sin(phi)$.

Another system behaving like a Foucault pendulum is a South Pointing Chariot that is run along a circle of fixed latitude on a globe. If the globe is not rotating in an inertial frame, the pointer on top of the chariot will indicate the direction of swing of a Foucault Pendulum that is traversing this latitude.

In physics, these systems are referred to as geometric phases. Mathematically they are understood through parallel transport.

Foucault pendula around the world

There are numerous Foucault pendula around the world, mainly at universities, science museums and planetariums. The experiment has even been carried out at the South Pole.

See also

• Geometric phase
• Parallel transport
• Coriolis force

References

• Persson, A. The Coriolis Effect: Four centuries of conflict between common sense and mathematics, Part I: A history to 1885 History of Meteorology 2 (2005)
• Phillips, N. A., What Makes the Foucault Pendulum Move among the Stars? Science and Education, Volume 13, Number 7, November 2004, pp. 653-661(9)
• Classical dynamics of particles and systems, 4ed, Marion Thornton (ISBN 0-03-097302-3 ), P.398-401.
• John B. Hart, Raymond E. Miller and Robert L. Mills A simple geometric model for visualizing the motion of a Foucault pendulum, American Journal of Physics -- January 1987 -- Volume 55, Issue 1, pp. 67-70
• Frank Wilczek and Alfred Shapere, "Geometric Phases in Physics", World Scientific, 1989
• Charles Wheatstone Wikisource: "Note relating to M. Foucault's new mechanical proof of the Rotation of the Earth", pp 65--68
• Julian Rubin, The Invention of the Foucault Pendulum, Following the Path of Discovery, 2007, retrieved 2007-10-31. Directions for repeating Foucault's experiment, on amateur science site.