![Panthéon Pendule de Foucault2](https://upload.wikimedia.org/wikipedia/commons/9/9c/Panth%C3%A9on_Pendule_de_Foucault2.JPG)
The Foucault pendulum or Foucault's pendulum is a simple device named after French physicist
Léon Foucault
Jean Bernard Léon Foucault (, ; ; 18 September 1819 – 11 February 1868) was a French physicist best known for his demonstration of the Foucault pendulum, a device demonstrating the effect of Earth's rotation. He also made an early measurement ...
, conceived as an experiment to demonstrate the
Earth's rotation
Earth's rotation or Earth's spin is the rotation of planet Earth around its own Rotation around a fixed axis, axis, as well as changes in the orientation (geometry), orientation of the rotation axis in space. Earth rotates eastward, in retrograd ...
. A long and heavy pendulum suspended from the high roof above a circular area was monitored over an extended time period, showing that the plane of oscillation rotated.
The
pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the ...
was introduced in 1851 and was the first experiment to give simple, direct evidence of the Earth's rotation. Foucault pendulums today are popular displays in
science museum
A science museum is a museum devoted primarily to science. Older science museums tended to concentrate on static displays of objects related to natural history, paleontology, geology, industry and industrial machinery, etc. Modern trends in mu ...
s and universities.
Original Foucault pendulum
![Foucault Pendulum](https://upload.wikimedia.org/wikipedia/commons/9/9c/Foucault_pendulum.jpg)
The first public exhibition of a Foucault pendulum took place in February 1851 in the Meridian of the
Paris Observatory
The Paris Observatory (french: Observatoire de Paris ), a research institution of the Paris Sciences et Lettres University, is the foremost astronomical observatory of France, and one of the largest astronomical centers in the world. Its histor ...
. A few weeks later, Foucault made his most famous pendulum when he suspended a brass-coated lead
bob
Bob, BOB, or B.O.B. may refer to:
Places
*Mount Bob, New York, United States
*Bob Island, Palmer Archipelago, Antarctica
People, fictional characters, and named animals
*Bob (given name), a list of people and fictional characters
*Bob (surname) ...
with a wire from the dome of the
Panthéon, Paris. The proper period of the pendulum was approximately
. Because the latitude of its location was
, the plane of the pendulum's swing made a full circle in approximately
, rotating clockwise approximately 11.3° per hour.
Focault explained his results in an 1851 paper entitled ''Physical demonstration of the Earth's rotational movement by means of the pendulum'', published in the ''
Weekly reports of the sessions of the Academy of Sciences''. He wrote that:
...an oscillatory movement of the pendulum mass follows an arc of a circle whose plane is well known, and to which the inertia
Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
of matter ensures an unchanging position in space. If these oscillations continue for a certain time, the movement of the earth, which continues to rotate from west to east, will become sensitive in contrast to the immobility of the oscillation plane whose trace on the ground will seem animated by a movement consistent with the apparent movement of the celestial sphere; and if the oscillations could be perpetuated for twenty-four hours, the trace of their plane would then execute an entire revolution around the vertical projection of the point of suspension.
The original bob used in 1851 at the Panthéon was moved in 1855 to the
Conservatoire des Arts et Métiers
A music school is an educational institution specialized in the study, training, and research of music. Such an institution can also be known as a school of music, music academy, music faculty, college of music, music department (of a larger ins ...
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 Métiers
The Musée des Arts et Métiers () ( French for Museum of Arts and Crafts) is an industrial design 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 p ...
before it reopened in 2000. On April 6, 2010, the cable suspending the bob in the Musée des Arts et Métiers snapped, causing irreparable damage to the pendulum bob and to the marble flooring of the museum. The original, now damaged pendulum bob is displayed in a separate case adjacent to the current pendulum display.
An exact copy of the original pendulum has been operating under the dome of the Panthéon, Paris since 1995.
Explanation of mechanics
![Foucault-rotz](https://upload.wikimedia.org/wikipedia/commons/8/82/Foucault-rotz.gif)
At either the
Geographic North Pole
The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is the point in the Northern Hemisphere where the Earth's axis of rotation meets its surface. It is called the True North Pole to distinguish from the Magn ...
or
Geographic South Pole
The South Pole, also known as the Geographic South Pole, Terrestrial South Pole or 90th Parallel South, is one of the two points where Earth's axis of rotation intersects its surface. It is the southernmost point on Earth and lies antipod ...
, the plane of oscillation of a pendulum remains fixed relative to the
distant masses of the universe while Earth rotates underneath it, taking one
sidereal day
Sidereal time (as a unit also sidereal day or sidereal rotation period) (sidereal ) is a timekeeping system that astronomers use to locate celestial objects. Using sidereal time, it is possible to easily point a telescope to the proper coord ...
to complete a rotation. So, relative to Earth, the plane of oscillation of a pendulum at the North Pole – viewed from above – undergoes a full clockwise rotation during one day; a pendulum at the South Pole rotates counterclockwise.
When a Foucault pendulum is suspended at the
equator
The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can als ...
, the plane of oscillation remains fixed relative to Earth. At other latitudes, the plane of oscillation
precesses relative to Earth, but more slowly than at the pole; the angular speed, (measured in clockwise
degrees per sidereal day), is proportional to the
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
of the
latitude
In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
, :
where latitudes north and south of the equator are defined as positive and negative, respectively. 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 latitude. For example, a Foucault pendulum at 30° south latitude, viewed from above by an earthbound observer, rotates counterclockwise 360° in two days.
Using enough wire length, the described circle can be wide enough that the tangential displacement along the measuring circle of between two oscillations can be visible by eye, rendering the Foucault pendulum a spectacular experiment: for example, the original Foucault pendulum in Panthéon moves circularly, with a 6-metre pendulum amplitude, by about 5 mm each period.
![Le petit Parisien illustre 2nov1902](https://upload.wikimedia.org/wikipedia/commons/2/23/Le_petit_Parisien_illustre_2nov1902.jpg)
A Foucault pendulum requires care to set up because imprecise construction can cause additional veering which masks the terrestrial effect. As observed by later Nobel laureate
Heike Kamerlingh Onnes
Heike Kamerlingh Onnes (21 September 1853 – 21 February 1926) was a Dutch physicist and Nobel laureate. He exploited the Hampson–Linde cycle to investigate how materials behave when cooled to nearly absolute zero and later to liquefy helium f ...
, who developed a fuller theory of the Foucault pendulum for his doctoral thesis (1879), geometrical imperfection of the system or elasticity of the support wire may cause an interference between two horizontal modes of oscillation, which caused Onnes' pendulum to go over from linear to elliptic oscillation in an hour. The initial launch of the pendulum is also 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 (see a
detail of the launch at the 50th anniversary in 1902).
Notably, veering of a pendulum was observed already in 1661 by
Vincenzo Viviani
Vincenzo Viviani (April 5, 1622 – September 22, 1703) was an Italian mathematician and scientist. He was a pupil of Torricelli and a disciple of Galileo. , a disciple of
Galileo
Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
, but there is no evidence that he connected the effect with the Earth's rotation; rather, he regarded it as a nuisance in his study that should be overcome with suspending the bob on two ropes instead of one.
Air resistance
In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...
damps the oscillation, so some Foucault pendulums 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. Besides air resistance (the use of a heavy symmetrical bob is to reduce friction forces, mainly air resistance by a symmetrical and aerodynamic bob) the other main engineering problem in creating a 1-meter Foucault pendulum nowadays is said to be ensuring there is no preferred direction of swing.
Precession as a form of parallel transport
![Parallel Transport](https://upload.wikimedia.org/wikipedia/commons/7/7d/Parallel_Transport.svg)
In a near-inertial frame moving in tandem with the Earth, but not sharing the rotation of the Earth about its own axis, the suspension point of the pendulum traces out a circular path during one sidereal day.
At the latitude of Paris, 48 degrees 51 minutes north, a full precession cycle takes just under 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 by just over 270 degrees. If the plane of swing was north–south at the outset, it is east–west one sidereal day later.
This also implies that there has been exchange of
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
; 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 an exchange must have occurred.
Rather than tracking the change of momentum, the precession of the oscillation plane can efficiently be described as a case of
parallel transport
In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection (vector bundle), c ...
. For that, it can be demonstrated, by composing the infinitesimal rotations, that the precession rate is proportional to the
projection
Projection, projections or projective may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphic ...
of the
angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
of Earth onto the
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
direction to Earth, which implies that the trace of the plane of oscillation will undergo parallel transport. After 24 hours, the difference between initial and final orientations of the trace in the Earth frame is , which corresponds to the value given by the
Gauss–Bonnet theorem
In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology.
In the simplest application, the case of a ...
. is also called the
holonomy
In differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus ...
or
geometric phase In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Ha ...
of the pendulum. When analyzing earthbound motions, the Earth frame is not an
inertial frame
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
, but rotates about the local vertical at an effective rate of radians per day. A simple method employing parallel transport 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 (the measuring circle and spectator are Earth-bounded, also if terrain reaction to Coriolis force is not perceived by spectator when he moves), using a rectangular coordinate system with its -axis pointing east and its -axis pointing north, the precession of the pendulum is due to the
Coriolis force
In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...
(other
fictitious forces
A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame.
It is related to Newton's second law of motion, which trea ...
as gravity and centrifugal force have not direct precession component, Euler's force is low because Earth's rotation speed is nearly constant). Consider a planar pendulum with constant natural frequency in the
small angle approximation
The small-angle approximations can be used to approximate the values of the main trigonometric functions, provided that the angle in question is small and is measured in radians:
:
\begin
\sin \theta &\approx \theta \\
\cos \theta &\approx 1 - \ ...
. There are two forces acting on the pendulum bob: the restoring force provided by gravity and the wire, and the Coriolis force (the centrifugal force, opposed to the gravitational restoring force, can be neglected). The Coriolis force at latitude is horizontal in the small angle approximation and is given by
where is the rotational frequency of Earth, is the component of the Coriolis force in the -direction and is the component of the Coriolis force in the -direction.
The restoring force, in the
small-angle approximation
The small-angle approximations can be used to approximate the values of the main trigonometric functions, provided that the angle in question is small and is measured in radians:
:
\begin
\sin \theta &\approx \theta \\
\cos \theta &\approx 1 - \ ...
and neglecting centrifugal force, is given by
![Foucault_pendulum_precession_vs_latitude](https://upload.wikimedia.org/wikipedia/commons/5/5d/Foucault_pendulum_precession_vs_latitude.svg)
Using
Newton's laws of motion
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in moti ...
this leads to the system of equations
Switching to complex coordinates , the equations read
To first order in this equation has the solution
If time is measured in days, then and the pendulum rotates by an angle of during one day.
Related physical systems
![Wheatstone Foucault device 256x256](https://upload.wikimedia.org/wikipedia/commons/d/d3/Wheatstone_Foucault_device_256x256.png)
Many physical systems precess in a similar manner to a Foucault pendulum. As early as 1836, the Scottish mathematician
Edward Sang
Edward Sang FRSE FRSSA LLD (30 January 1805 – 23 December 1890) was a Scottish mathematician and civil engineer, best known for having computed large tables of logarithms, with the help of two of his daughters. These tables went beyond the tab ...
contrived and explained the precession of a spinnin
top In 1851,
Charles Wheatstone
Sir Charles Wheatstone FRS FRSE DCL LLD (6 February 1802 – 19 October 1875), was an English scientist and inventor of many scientific breakthroughs of the Victorian era, including the English concertina, the stereoscope (a device for di ...
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 nonspinning, 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 .
Foucault-like precession is observed in a virtual system wherein a massless particle is constrained to remain on a rotating plane that is inclined with respect to the axis of rotation.
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
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inert ...
can be treated as a sphere ''S''
3 in 4-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
with imaginary radius and imaginary timelike coordinate. Parallel transport of polarization vectors along such sphere gives rise to
Thomas precession
In physics, the Thomas precession, named after Llewellyn Thomas, is a 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 pa ...
, which is analogous to the rotation of the swing plane of Foucault pendulum due to parallel transport along a sphere ''S''
2 in 3-dimensional Euclidean space.
In physics, the evolution of such systems is determined by
geometric phase In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Ha ...
s. Mathematically they are understood through parallel transport.
Foucault pendulums around the world
There are numerous Foucault pendulums at universities, science museums, and the like throughout the world. The
United Nations General Assembly Building
The United Nations General Assembly Building is part of the headquarters of the United Nations in the Turtle Bay, Manhattan, Turtle Bay neighborhood of Manhattan in New York City. It contains the main assembly hall of the United Nations General ...
at the
United Nations headquarters
The United Nations is headquartered in Midtown Manhattan, New York City, United States, and the complex has served as the official headquarters of the United Nations since its completion in 1951. It is in the Turtle Bay, Manhattan, Turtle Bay neig ...
in New York City has one. The
Oregon Convention Center
The Oregon Convention Center is a convention center in Portland, Oregon. Completed in 1989 and opened in 1990, it is located on the east side of the Willamette River in the Lloyd District neighborhood. It is best known for the twin spire towers, ...
pendulum is claimed to be the largest, its length is approximately , however, there are larger ones listed in the article, such as the one in Gamow Tower at the University of Colorado (39.3 m). There used to be much longer pendulums, such as the pendulum in
Saint Isaac's Cathedral
Saint Isaac's Cathedral or Isaakievskiy Sobor (russian: Исаа́киевский Собо́р) is a large architectural landmark cathedral that currently functions as a museum with occasional church services in Saint Petersburg, Russia. It is ...
,
Saint Petersburg
Saint Petersburg ( rus, links=no, Санкт-Петербург, a=Ru-Sankt Peterburg Leningrad Petrograd Piter.ogg, r=Sankt-Peterburg, p=ˈsankt pʲɪtʲɪrˈburk), formerly known as Petrograd (1914–1924) and later Leningrad (1924–1991), i ...
,
Russia
Russia (, , ), or the Russian Federation, is a List of transcontinental countries, transcontinental country spanning Eastern Europe and North Asia, Northern Asia. It is the List of countries and dependencies by area, largest country in the ...
.
File:Musée des Arts et Métiers 7, Paris.JPG, Foucault pendulum at the Musée des Arts et Métiers
The Musée des Arts et Métiers () ( French for Museum of Arts and Crafts) is an industrial design 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 p ...
File:Foucault Pendulum - Ranchi Science Centre - Jharkhand 2010-11-29 8871.JPG, Foucault pendulum at the Ranchi Science Centre
Ranchi Science Centre is the first science centre in the state of Jharkhand in India, under Jharkhand Council on Science & Technology (JCST) of Department of Science & Technology, Government of Jharkhand (GOJ). The centre comprises two storied bu ...
File:California Academy of Sciences Foucault Pendulum Clock.jpg, Foucault pendulum at the California Academy of Sciences
The California Academy of Sciences is a research institute and natural history museum in San Francisco, California, that is among the largest museums of natural history in the world, housing over 46 million specimens. The Academy began in 1853 ...
File:Foucault Pendulum at Buxton - geograph.org.uk - 1769680.jpg, Foucault pendulum at the Devonshire Dome
The Devonshire Dome building (previously known as the Devonshire Royal Hospital) is a Listed building, Grade II* listed 18th-century former Stable, stable block in Buxton, Derbyshire. It was built by John Carr (architect), John Carr of York and ...
, University of Derby
, mottoeng = Experience is the best teacher
, established = 1851 – Teacher Training College1992 – gained university status
, type = Public
, chancellor = William Cavendish, Ear ...
South Pole
The experiment has also been carried out at the
South Pole
The South Pole, also known as the Geographic South Pole, Terrestrial South Pole or 90th Parallel South, is one of the two points where Earth's axis of rotation intersects its surface. It is the southernmost point on Earth and lies antipod ...
, where it was assumed that the rotation of the Earth would have maximum effect.
A pendulum was installed in a six-story staircase of a new station under construction at the
Amundsen-Scott South Pole Station. It had a length of and the bob weighed . The location was ideal: no moving air could disturb the pendulum. The researchers confirmed about 24 hours as the rotation period of the plane of oscillation.
See also
*
Absolute rotation
In physics, the concept of absolute rotation— rotation independent of any external reference—is a topic of debate about relativity, cosmology, and the nature of physical laws.
For the concept of absolute rotation to be scientifically meaning ...
*
Coriolis effect
In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...
*
Earth's rotation
Earth's rotation or Earth's spin is the rotation of planet Earth around its own Rotation around a fixed axis, axis, as well as changes in the orientation (geometry), orientation of the rotation axis in space. Earth rotates eastward, in retrograd ...
*
Eötvös experiment
The Eötvös experiment was a famous physics experiment that measured the correlation between inertial mass and gravitational mass, demonstrating that the two were one and the same, something that had long been suspected but never demonstrated with ...
*
Inertial frame
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
*
Lariat chain
A Lariat chain is a loop of chain that hangs off, and is spun by a wheel. It is often used as a science exhibit or a toy.
The original Lariat Chain was created in 1986 by Norman Tuck, as an Artist-in-Residence project at the Exploratorium in Sa ...
*
Precession
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In othe ...
References
Further reading
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*
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External links
* Wolfe, Joe,
A derivation of the precession of the Foucault pendulum.
*
The Foucault Pendulum, derivation of the precession in polar coordinates.
*
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 an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
: a computer model of the pendulum allowing manipulation of pendulum frequency, Earth rotation frequency, latitude, and time.
*
Webcam Kirchhoff-Institut für Physik, Universität Heidelberg.
Foucault pendulum explanation, in friendly format
Exposition including a tabletop device that shows the Foucault effect in seconds.
* Foucault, M. L.
Franklin Institute, 2000, retrieved 2007-10-31. Translation of his paper on Foucault pendulum.
*
*
Pendolo nel SaloneThe Foucault Pendulum inside Palazzo della Ragione in Padova, Italy
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{{Authority control
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