# Foucault pendulum

The

**Foucault pendulum**(pronEng|fuːˈkoʊ "foo-KOH"), or**Foucault's pendulum**, named after the French physicistLéon Foucault , was conceived as an experiment to demonstrate the rotation of the Earth.**The experiment**The experimental apparatus consists of a tall

pendulum free tooscillate 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 February1851 in the Meridian Room of theParis 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 inParis . 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 orSouth Pole , the plane of oscillation of a pendulum remains fixed with respect to thefixed stars while Earth rotates underneath it, taking onesidereal 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 theequator , 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 persidereal day ), is proportional to thesine of thelatitude , $phi$::$alpha=360,sin(phi).$

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**[

angle persidereal day as a function oflatitude . The pendulum rotates in the anticlockwise (positive) direction on the southern hemisphere and in the clockwise (negative) direction on the northern hemisphere. The only points where the pendulum returns to its original orientation after one day are the poles and the equator.]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 undergoesparallel transport . The difference between initial and final orientations is $alpha=-2pi,sin(phi)$ as given by theGauss-Bonnet theorem . $alpha$ is also called theholonomy orgeometric phase of the pendulum. Thus, when analyzing earthbound motions, the Earth frame is not aninertial 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 theangular 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 thesmall angle approximation . There are two forces acting on the pendulum bob: the restoring force provided by gravity and the wire, and theCoriolis force . The Coriolis force atlatitude $phi$ is horizontal in the small angle approximation and is given by:$egin\{align\}F\_\{c,x\}\; =\; 2\; m\; Omega\; dfrac\{dy\}\{dt\}\; sin(phi)\backslash \backslash F\_\{c,y\}\; =\; -\; 2\; m\; Omega\; dfrac\{dx\}\{dt\}\; sin(phi)end\{align\}$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:$egin\{align\}F\_\{g,x\}\; =\; -\; m\; omega^2\; x\; \backslash \backslash F\_\{g,y\}\; =\; -\; m\; omega^2\; y.end\{align\}$

[southern hemisphere . The rate of rotation is greatly exaggerated. A real Foucault pendulum, released from rest, does not pass directly over its equilibrium position as the one in the animation does.] UsingNewton's laws of motion this leads to the system of equations:$egin\{align\}dfrac\{d^2x\}\{dt^2\}\; =\; -omega^2\; x\; +\; 2\; Omega\; dfrac\{dy\}\{dt\}\; sin(phi)\backslash \backslash dfrac\{d^2y\}\{dt^2\}\; =\; -omega^2\; y\; -\; 2\; Omega\; dfrac\{dx\}\{dt\}\; sin(phi).end\{align\}$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:$z=e^\{-iOmega\; sin(phi)t\}left(c\_1\; e^\{iomega\; t\}+c\_2\; e^\{-iomega\; t\}\; ight).$

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 phase s. Mathematically they are understood throughparallel 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.

**ee also***

Geometric phase

*Parallel transport

*Coriolis force **References*** Persson, A. " [

*http://www.meteohistory.org/2005historyofmeteorology2/01persson.pdf The Coriolis Effect: Four centuries of conflict between common sense and mathematics, Part I: A history to 1885*] " History of Meteorology 2 (2005)* [

*http://www.fi.edu/tfi/exhibits/bower/03/earth.html 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 [

*http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=AJPIAS000055000001000067000001&idtype=cvips&ident=freesearch&prog=search 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: "", pp 65--68*Julian Rubin, [

*http://www.juliantrubin.com/bigten/foucaultpendulum.html "The Invention of the Foucault Pendulum"*] , Following the Path of Discovery, 2007, retrieved2007-10-31 . Directions for repeating Foucault's experiment, on amateur science site.**External links****Derivations**

* Wolfe, Joe, " [*http://www.phys.unsw.edu.au/~jw/pendulumdetails.html A derivation of the precession of the Foucault pendulum*] ".

* " [*http://www.sciencebits.com/foucault The Foucault Pendulum*] ", derivation of the precession in polar coordinates.**Visualisations, video imaging and models**

* " [*http://www.phys.unsw.edu.au/PHYSICS_!/FOUCAULT_PENDULUM/foucault_pendulum.html The Foucault Pendulum*] " By Joe Wolfe, with film clip and animations.

* " [*http://demonstrations.wolfram.com/FoucaultsPendulum/ Foucault's Pendulum*] " by Jens-Peer Kuska with Jeff Bryant,The Wolfram Demonstrations Project : a computer model of the pendulum allowing manipulation of pendulum frequency, Earth rotation frequency, latitude, and time.

* " [*http://pendelcam.kip.uni-heidelberg.de/ Webcam Kirchhoff-Institut für Physik, Universität Heidelberg*] ".

* " [*http://www.calacademy.org/products/pendulum/index.html California academy of sciences, CA*] " Foucault pendulum explanation, in friendly format

* " [*http://www.physics.buffalo.edu/ubexpo/listofexhibits/Foucault.html Foucault pendulum model*] " Exposition including a tabletop device that shows the Foucault effect in seconds.**History**

* Foucault, M. L., [*http://www.fi.edu/time/journey/Pendulum/foucault_paper_page_one.html "Physical demonstration of the rotation of the Earth by means of the pendulum"*] , Franklin Institute, 2000, retrieved2007-10-31 . Translation of his paper on Foucault pendulum.

* Tobin, William " [*http://www2.phys.canterbury.ac.nz/~wjt23/foucault.html The Life and Science of Léon Foucault*] ".**Notable**

* " [*http://www.phys-astro.sonoma.edu/people/students/baker/SouthPoleFoucault.html South Pole Foucault Pendulum*] ". Winter, 2001.**Educational supplies**

* " [*http://www.science-design.com Science Design's Small Portable Foucault Pendulum*] " A company selling a Foucault Pendulum for the classroom.

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Foucault pendulum**— n. a pendulum consisting of a heavy weight on the end of a long wire hanging from a fixed point, of the kind invented by Jean Foucault to show that the earth is rotating: the pattern of the pendulum s swinging appears to be rotating to an… … English World dictionary**Foucault pendulum**— Physics. a pendulum that demonstrates the rotation of the earth by exhibiting an apparent change in its plane of oscillation. [1850 55; named after J.B.L. FOUCAULT] * * * Large pendulum that is free to swing in any direction. As it swings back… … Universalium**Foucault pendulum**— Fuko švytuoklė statusas T sritis fizika atitikmenys: angl. Foucault pendulum vok. Foucaultsches Pendel, n rus. маятник Фуко, m pranc. pendule de Foucault, m … Fizikos terminų žodynas**Foucault pendulum**— noun pendulum with a long wire; can swing in any direction; the change in the swing plane demonstrates the earth s rotation • Hypernyms: ↑pendulum * * * noun Usage: usually capitalized F Etymology: after J.B.L. Foucault : a freely swinging… … Useful english dictionary**Foucault pendulum**— noun Etymology: J.B.L. Foucault Date: 1931 a freely swinging pendulum that consists of a heavy weight hung by a long wire and that swings in a constant direction which appears to change showing that the earth rotates … New Collegiate Dictionary**Foucault pendulum**— For the device (pronounced foo ko), but Foucault’s Pendulum (1988) for the novel by Umberto Eco … Bryson’s dictionary for writers and editors**Foucault pendulum**— /fukoʊ ˈpɛndʒələm/ (say foohkoh penjuhluhm) noun a long simple pendulum which demonstrates the rotation of the earth by changing its plane of oscillation during the course of a day. {demonstrated by Jean Bernard Léon Foucault, 1819–68, French… … Australian English dictionary**Foucault pendulum**— noun Foucaults pendulum … Wiktionary**Foucault Pendulum vector diagrams**— Several vector diagrams are often used to demonstrate the physics underlying the Foucault pendulum.Diagrams are provided to illustrate a pendulum located at the North Pole, equator, and 45 degrees N to show how the rotation of Earth in relation… … Wikipedia**Foucault pendulum vector diagrams**— Several vector diagrams are often used to demonstrate the physics underlying the Foucault pendulum. Diagrams are provided to illustrate a pendulum located at the North Pole, equator, and 45 degrees N to show how the rotation of Earth in relation… … Wikipedia