NationMaster - Encyclopedia: Foucault pendulum

The Foucault pendulum (pronounced "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. Foucaults Pendulum, The Pantheon, Paris. ... The PanthÃ©on Interior Dome of the PanthÃ©on Entrance of the PanthÃ©on Voltaires statue and tomb in the crypt of the PanthÃ©on The PanthÃ©on (Latin Pantheon[1], from Greek Pantheon, meaning All the Gods) is a building in the Latin Quarter in Paris, France. ... ... This article is about Earth as a planet. ...

1 The experiment
2 The dynamics of the Foucault pendulum
3 Related physical systems
4 Foucault pendula in the world
5 See also
6 External links
7 References

The experiment

It is 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. Simple gravity pendulum assumes no air resistance and no friction of/at the nail/screw. ... 1851 (MDCCCLI) was a common year starting on Wednesday (see link for calendar) of the Gregorian calendar (or a common year starting on Friday of the 12-day-slower Julian calendar). ... gros pd]], enfoire at NanÃ§ay. ... A bob is the weight on the end of a pendulum. ... The PanthÃ©on Interior Dome of the PanthÃ©on Entrance of the PanthÃ©on Voltaires statue and tomb in the crypt of the PanthÃ©on The PanthÃ©on (Latin Pantheon[1], from Greek Pantheon, meaning All the Gods) is a building in the Latin Quarter in Paris, France. ... City flag City coat of arms Motto: Fluctuat nec mergitur (Latin: Tossed by the waves, she does not sink) The Eiffel Tower in Paris, as seen from the esplanade du TrocadÃ©ro. ...

A Foucault pendulum at the north pole. The pendulum swings in the same plane as the Earth rotates beneath it.

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. Image File history File links Foucault_pendulum_at_north_pole_accurate. ... Image File history File links Foucault_pendulum_at_north_pole_accurate. ...

Animation of a Foucault pendulum, with the rotation rate greatly exaggerated. The green trace shows the path of the pendulum bob over the ground, and the blue trace shows the path in a frame of reference rotating with the plane of the pendulum.

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 undergoes a full clockwise or anticlockwise 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 plane of oscillation of a pendulum with respect to Earth rotates with an angular speed proportional to the sine of its latitude; for example, one at 30° latitude rotates 180° in one day. The general formula for $α$ , the angle of rotation (measured in a counterclockwise direction) that the pendulum undergoes during one day at latitude $φ$ , is Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... For other uses, see North Pole (disambiguation). ... For other uses, see South Pole (disambiguation). ... On a prograde planet like the Earth, the sidereal day is shorter than the solar day. ... World map showing the equator in red In tourist areas, the equator is often marked on the sides of roads The equator marked as it crosses IlhÃ©u das Rolas, in SÃ£o TomÃ© and PrÃncipe. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... Latitude,usually denoted symbolically by the Greek letter phi, , gives the location of a place on Earth north or south of the equator. ... An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ...

$alpha=-2pi,sin(phi)$ .

Many people found the sine factor difficult to understand, which prompted Foucault to conceive the gyroscope in 1852. The gyroscope's spinning rotor tracks the stars directly. Its axis of rotation returns to its original orientation after one day whatever the latitude, unaffected by the sine factor. In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... A gyroscope For other uses, see Gyroscope (disambiguation). ...

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, without imparting any unwanted sideways motion, is to use a flame to burn through a thread which is temporarily holding the bob in its starting position. Air resistance damps the oscillation, so Foucault pendulums in museums usually incorporate an electromagnetic or other drive to keep the bob swinging or are restarted regularly. In the latter case, a launching ceremony may be performed as an added show. For a solid object moving through a fluid or gas, drag is the sum of all the aerodynamic or hydrodynamic forces in the direction of the external fluid flow. ...

The dynamics of the Foucault pendulum

Change of direction of the plane of swing of the pendulum in angle per sidereal day as a function of latitude. 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 undergoes parallel transport. The difference between initial and final orientations is $alpha=-2pi,sin(phi)$ as given by the Gauss-Bonnet theorem. $α$ 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. Image File history File links No higher resolution available. ... An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ... On a prograde planet like the Earth, the sidereal day is shorter than the solar day. ... Latitude,usually denoted symbolically by the Greek letter phi, , gives the location of a place on Earth north or south of the equator. ... On a prograde planet like the Earth, the sidereal day is shorter than the solar day. ... Constraint is an equation that defines a restriction of solutions of an optimization problem to a so called feasible set. ... In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ... The Gauss-Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). ... In differential geometry, the holonomy group of a connection on a vector bundle over a smooth manifold M is the group of linear transformations induced by parallel transport around closed loops in M. There is an analogous notion for connections on principal bundles over M. The holonomy group of a... In quantum mechanics, the Berry phase is a phase acquired by quantum states when subjected to adiabatic processes, resulting from the geometrical properties of the parameter space of the Hamiltonian. ... In physics, an inertial frame of reference, or inertial frame for short (also descibed as absolute frame of reference), is a frame of reference in which the observers move without the influence of any accelerating or decelerating force. ... The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs. ... In geometry, an orthogonal projection of a k-dimensional object onto a d-dimensional hyperplane (d < k) is obtained by intersections of (k − d)- dimensional hyperplanes drawn through the points of an object orthogonally to the d-hyperplane. ... Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ... A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ...

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 $ω$ 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 $φ$ is horizontal in the small angle approximation and is given by In physics, the Coriolis effect is an inertial force first described by Gaspard-Gustave Coriolis, a French scientist, in 1835. ... It has been suggested that this article or section be merged with Small-angle formula. ... In physics, the Coriolis effect is an inertial force first described by Gaspard-Gustave Coriolis, a French scientist, in 1835. ... Latitude,usually denoted symbolically by the Greek letter phi, , gives the location of a place on Earth north or south of the equator. ...

$begin{align} F_{c,x} &= 2 m Omega dfrac{dy}{dt} sin(phi) F_{c,y} &= - 2 m Omega dfrac{dx}{dt} sin(phi) end{align}$

where $Ω$ 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 Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

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

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

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

To first order in $Ω / ω$ 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 $Ω = 2π$ and we see that the pendulum rotates by an angle of $-2pi,sin(phi)$ during one day.

Related physical systems

The device described by Wheatstone.

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 string that is mounted on top of a disk so that it makes a fixed angle $φ$ with the disk. The string 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 $φ$ . Image File history File links No higher resolution available. ... Charles Wheatstone Sir Charles Wheatstone (February 6, 1802 - October 19, 1875) was the British inventor of many innovations including the English concertina the Stereoscope an early form of microphone the Playfair cipher (named for Lord Playfair, the person who publicized it) He was a major figure in the development of...

Similarly, consider a non-spinning perfectly balanced bicycle wheel mounted on a disk so that its axis of rotation makes an angle $φ$ 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. South Pointing Chariot (replica) Supposedly invented sometime around 2600BC in China by the Yellow Emperor Huang Di, the South Pointing Chariot (Zhi Nan Ju æŒ‡å—è»Š) is widely regarded as the most complex geared mechanism of the ancient Chinese civilization. ...

In physics, these systems are referred to as geometric phases. Mathematically they are understood through parallel transport. In quantum mechanics, the Berry phase is a phase acquired by quantum states when subjected to adiabatic processes, resulting from the geometrical properties of the parameter space of the Hamiltonian. ... In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ...

The animation describes the motion of a Foucault Pendulum at a latitude of 30°N. The plane of oscillation rotates by an angle of -180° during one day, so after two days the plane returns to its original orientation.

Image File history File links No higher resolution available. ...

Foucault pendula in the world

Further information: List of Foucault pendula

There is an abundance of Foucault pendula in the world, mainly at universities, science museums and planetariums. The experiment has even been carried out at the South Pole. This is a list of Foucault pendula in the world: // Foucaults Pendulum in the PanthÃ©on, Paris. ...

External links

"South Pole Foucault Pendulum". Winter, 2001.
Wolfe, Joe, "A derivation of the precession of the Foucault pendulum".
"The Foucault Pendulum", derivation of the precession in polar coordinates.
"Webcam Kirchhoff-Institut für Physik, Universität Heidelberg".
Tobin, William "The Life and Science of Léon Foucault".
Lansey, Jonathan "Bowling Ball Foucault's Pendulum Experiment".
"The Foucault Pendulum" with film clip and animations.
"Foucault pendulum video" Foucault pendulum at the Musée des arts et métiers, Paris, France) (video clip)
"Science Design's Small Portable Foucault Pendulum" A company selling a Foucault Pendulum for the classroom.
"Foucault pendulum at Lexington Public Library in Lexington, KY " Foucault pendulum at Lexington Public Library in Lexington, KY

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

Category: Pendulums Frank Wilczek (born May 15, 1951) is a Nobel prize winning American physicist. ... Charles Wheatstone Sir Charles Wheatstone (February 6, 1802 - October 19, 1875) was the British inventor of many innovations including the English concertina the Stereoscope an early form of microphone the Playfair cipher (named for Lord Playfair, the person who publicized it) He was a major figure in the development of...

Results from FactBites:

Eco - Reviews: Humanitas review of "Foucault's Pendulum" (1632 words)
	Foucault's Pendulum will almost certainly achieve recognition as well, for it is a complex artifact of Eco's postmodern aesthetic at work in a traditional literary form: in this case like his first novel, the detective thriller.
	The poetry of the pendulum is the poetry of Eco's novel, and of history itself.
	Belbo is killed, magnificently, symbolically, hung by the wire of the pendulum.

Foucault pendulum - Wikipedia, the free encyclopedia (2780 words)
	Foucault pendulum at the Musée des arts et métiers (Paris); pegs are placed around and are knocked down as the pendulum swing plane veers.
	The Foucault pendulum that hangs in the rotunda of the Lexington Public Library in Lexington, Kentucky in the United States is the largest ceiling clock in the world.
	The origin of the force that is involved in the precession of the Foucault pendulum with respect to the fixed stars is the centripetal force that keeps objects on the Earth's surface moving in circular paths.

More results at FactBites »

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