The
Foucault pendulum or
Foucault's pendulum, named after the French physicist
Léon FoucaultJean Bernard Léon Foucault was a French physicist best known for the invention of the Foucault pendulum, a device demonstrating the effect of the Earth's rotation...
, is a simple device conceived as an experiment to demonstrate the rotation of the Earth. While it had long been known that the Earth rotated, the introduction of the Foucault pendulum in 1851 was the first simple proof of the rotation in an easy-to-see experiment. Today, Foucault pendula are popular displays in science museums and universities.
Experiment
The experimental apparatus consists of a tall
pendulumA pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position...
free to swing in any vertical plane. The actual plane of swing appears to rotate relative to the Earth; in fact the plane is fixed in space while the Earth rotates under the pendulum once a sidereal day. The first public exhibition of a Foucault pendulum took place in February 1851 in the Meridian of the
Paris ObservatoryThe Paris Observatory is the foremost astronomical observatory of France, and one of the largest astronomical centres in the world...
. A few weeks later Foucault made his most famous pendulum when he suspended a 28 kg brass-coated lead
bobA bob is the weight on the end of a pendulum most commonly, but not exclusively, found in pendulum clocks.- Reason for use :Although a pendulum can theoretically be any shape, any rigid object swinging on a pivot, clock pendulums are usually made of a weight or bob attached to the bottom end of a...
with a 67 meter long wire from the dome of the
Panthéon, ParisThe Panthéon is a building in the Latin Quarter in Paris. It was originally built as a church dedicated to St. Genevieve and to house the reliquary châsse containing her relics but, after many changes, now functions as a secular mausoleum containing the remains of distinguished French citizens...
. The plane of the pendulum's swing rotated clockwise 11° per hour, making a full circle in 32.7 hours. The original bob used in 1851 at the Panthéon was moved in 1855 to the Conservatoire des Arts et Métiers in Paris. A second temporary installation was made for the 50th anniversary in 1902.
During museum reconstruction in the 1990s the original pendulum was temporarily displayed at the Panthéon (1995), but was later returned to the
Musée des Arts et MétiersThe Musée des Arts et Métiers is a museum in Paris that houses the collection of the Conservatoire National des Arts et Métiers , which was founded in 1794 as a repository for the preservation of scientific instruments and inventions.-History:Since its foundation, the museum has been housed in the...
. On April 6, 2010, the cable suspending the bob in the
Musée des Arts et MétiersThe Musée des Arts et Métiers is a museum in Paris that houses the collection of the Conservatoire National des Arts et Métiers , which was founded in 1794 as a repository for the preservation of scientific instruments and inventions.-History:Since its foundation, the museum has been housed in the...
snapped causing irreparable damage to the pendulum and to the marble flooring of the museum. An exact copy of the original pendulum has been swinging permanently since 1995 under the dome of the
Panthéon, ParisThe Panthéon is a building in the Latin Quarter in Paris. It was originally built as a church dedicated to St. Genevieve and to house the reliquary châsse containing her relics but, after many changes, now functions as a secular mausoleum containing the remains of distinguished French citizens...
.
At either the
North PoleThe North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is, subject to the caveats explained below, defined as the point in the northern hemisphere where the Earth's axis of rotation meets its surface...
or
South PoleThe South Pole, also known as the Geographic South Pole or Terrestrial South Pole, is one of the two points where the Earth's axis of rotation intersects its surface. It is the southernmost point on the surface of the Earth and lies on the opposite side of the Earth from the North Pole...
, the plane of oscillation of a pendulum remains fixed relative to the
distant masses of the universeIn theoretical physics, particularly in discussions of gravitation theories, Mach's principle is the name given by Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach....
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 Pole undergoes a full clockwise rotation during one day; a pendulum at the South Pole rotates counterclockwise.
When a Foucault pendulum is suspended at the
equatorAn equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....
, 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,
ω (measured in clockwise
degreesA degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...
per sidereal day), is proportional to the
sineIn mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
of the
latitudeIn geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
,
φ:
where 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 360° in two days.
In order to demonstrate the rotation of the Earth without the complication of the dependence on latitude, Foucault used a
gyroscopeA gyroscope is a device for measuring or maintaining orientation, based on the principles of angular momentum. In essence, a mechanical gyroscope is a spinning wheel or disk whose axle is free to take any orientation...
in an 1852 experiment. 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, not subject to the unbalanced
CoriolisIn physics, the Coriolis effect is a deflection of moving objects when they are viewed in a rotating reference frame. In a reference frame with clockwise rotation, the deflection is to the left of the motion of the object; in one with counter-clockwise rotation, the deflection is to the right...
forces acting on the pendulum as a result of its geometric asymmetry.
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 some Foucault pendula in museums incorporate an electromagnetic or other drive to keep the bob swinging; others are restarted regularly, sometimes with a launching ceremony as an added attraction.
A
pendulum day is the time needed for the plane of a freely suspended Foucault pendulum to complete an apparent rotation about the local vertical. This is one sidereal day divided by the sine of the
latitudeIn geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
.
Precession as a form of parallel transport
From the perspective of an inertial frame moving in tandem with Earth, but not sharing its rotation, the suspension point of the pendulum traces out a circular path during one sidereal day. At the latitude of Paris a full precession cycle takes 32 hours, so after one sidereal day, when the Earth is back in the same orientation as one sidereal day before, the oscillation plane has turned 90 degrees. If the plane of swing was north-south at the outset, it is east-west one sidereal day later. This implies that there has been exchange of momentum; the Earth and the pendulum bob have exchanged momentum. (The Earth is so much more massive than the pendulum bob that the Earth's change of momentum is unnoticeable. Nonetheless, since the pendulum bob's plane of swing has shifted the conservation laws imply that there must have been exchange.)
Rather than tracking the change of momentum the precession of the oscillation plane can efficiently be described as a case of
parallel transportIn geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
. For that it is assumed that the precession rate is proportional to the projection of the
angular velocityIn physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
of Earth onto the
normalA surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...
direction to Earth, which implies that the plane of oscillation will undergo parallel transport. The difference between initial and final orientations is , in which case the Gauss-Bonnet theorem applies.
α is also called the
holonomyIn differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections,...
or
geometric phaseIn classical and quantum mechanics, the geometric phase, Pancharatnam–Berry phase , Pancharatnam phase or most commonly Berry phase, is a phase acquired over...
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 radians per day.
A simple method employing
parallel transportIn geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
within cones tangent to the Earth's surface can be used to describe
the rotation angle of the swing plane of Foucault's pendulum.
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 described by the Coriolis force. Consider a planar pendulum with natural frequency
ω in the
small angle approximationThe small-angle approximation is a useful simplification of the basic trigonometric functions which is approximately true in the limit where the angle approaches zero. They are truncations of the Taylor series for the basic trigonometric functions to a second-order 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
latitudeIn geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
φ is horizontal in the small angle approximation and is given by
where Ω is the rotational frequency of Earth,
Fc,
x is the component of the Coriolis force in the
x-direction and
Fc,
y is the component of the Coriolis force in the
y-direction.
The restoring force, in the small angle approximation, is given by
Using
Newton's laws of motionNewton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...
this leads to the system of equations
Switching to complex coordinates , the equations read
To first order in Ω/
ω this equation has the solution
If we measure time in days, then and we see that the pendulum rotates by an angle of −2π sin(
φ) during one day.
Related physical systems
There are many physical systems that precess in a similar manner to a Foucault pendulum. In 1851,
Charles WheatstoneSir Charles Wheatstone FRS , was an English scientist and inventor of many scientific breakthroughs of the Victorian era, including the English concertina, the stereoscope , and the Playfair cipher...
described an apparatus that consists of a vibrating spring that is mounted on top of a disk so that it makes a fixed angle
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
.
Similarly, consider a non-spinning perfectly balanced bicycle wheel mounted on a disk so that its axis of rotation makes an angle
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
.
Another system behaving like a Foucault pendulum is a
South Pointing ChariotThe south-pointing chariot was an ancient Chinese two-wheeled vehicle that carried a movable pointer to indicate the south, no matter how the chariot turned. Usually, the pointer took the form of a doll or figure with an outstretched arm...
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.
Spin of a relativistic particle moving in a circular orbit precesses similar to the swing plane of Foucault pendulum. The relativistic velocity space in Minkowski spacetime can be treated as a sphere
S3 in 4-dimensional
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
with imaginary radius and imaginary timelike coordinate.
Parallel transportIn geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
of polarization vectors along such sphere gives rise to
Thomas precessionIn physics the Thomas precession, named after Llewellyn Thomas, is a special relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope and relates the angular velocity of the spin of a particle following a curvilinear orbit to the angular...
, which is analogous to the rotation of the swing plane of Foucault pendulum due to parallel transport along a sphere
S2 in 3-dimensional Euclidean space.
In physics, the evolution of such systems is determined by
geometric phaseIn classical and quantum mechanics, the geometric phase, Pancharatnam–Berry phase , Pancharatnam phase or most commonly Berry phase, is a phase acquired over...
s. Mathematically they are understood through
parallel transportIn geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
.
Foucault pendula around the world
There are numerous Foucault pendula around the world, mainly at universities, science museums and planetaria. A particularly famous and prominent one is located at the
United NationsThe United Nations is an international organization whose stated aims are facilitating cooperation in international law, international security, economic development, social progress, human rights, and achievement of world peace...
in
ManhattanManhattan is the oldest and the most densely populated of the five boroughs of New York City. Located primarily on the island of Manhattan at the mouth of the Hudson River, the boundaries of the borough are identical to those of New York County, an original county of the state of New York...
. The experiment has been carried out at the South Pole.
See also
- Coriolis force
- Geometric phase
In classical and quantum mechanics, the geometric phase, Pancharatnam–Berry phase , Pancharatnam phase or most commonly Berry phase, is a phase acquired over...
- Hannay angle
In classical mechanics, the Hannay angle is a mechanics analogue of the Berry phase. It was named after John Hannay of the University of Bristol, UK.-Example:...
- Rose curve, the shape traced by the bob as seen from directly above
- Foucault's Pendulum
Foucault's Pendulum is a novel by Italian writer and philosopher Umberto Eco. It was first published in 1988; the translation into English by William Weaver appeared a year later....
– a novel by Italian writer and philosopher Umberto EcoUmberto Eco Knight Grand Cross is an Italian semiotician, essayist, philosopher, literary critic, and novelist, best known for his novel The Name of the Rose , an intellectual mystery combining semiotics in fiction, biblical analysis, medieval studies and literary theory...
.
External links and bibliography
- 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.
- V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag (1989), ISBN 0-387-96890-3, p. 123
- 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.
- Wolfe, Joe, "A derivation of the precession of the Foucault pendulum".
- "The Foucault Pendulum", derivation of the precession in polar coordinates.
- "The Foucault Pendulum" By Joe Wolfe, with film clip and animations.
- "Foucault's Pendulum" by Jens-Peer Kuska with Jeff Bryant, Wolfram Demonstrations Project
The Wolfram Demonstrations Project is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. It consists of an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and...
: a computer model of the pendulum allowing manipulation of pendulum frequency, Earth rotation frequency, latitude, and time.
- "Webcam Kirchhoff-Institut für Physik, Universität Heidelberg".
- California academy of sciences, CA Foucault pendulum explanation, in friendly format
- Foucault pendulum model Exposition including a tabletop device that shows the Foucault effect in seconds.
- Foucault, M. L., Physical demonstration of the rotation of the Earth by means of the pendulum, Franklin Institute, 2000, retrieved 2007-10-31. Translation of his paper on Foucault pendulum.
- Tobin, William "The Life and Science of Léon Foucault".
- "South Pole Foucault Pendulum". Winter, 2001.