Editing Foucault pendulum

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{{Short description|Device to demonstrate Earth's rotation}}
{{About|the physics experiment and instrument|the novel by Umberto Eco|Foucault's Pendulum}}
[[File:Panthéon Pendule de Foucault2.JPG|thumb|upright=1.15|Foucault's pendulum in the [[Panthéon]], [[Paris]]]]
The '''Foucault pendulum''' or '''Foucault's pendulum''' is a simple device named after French physicist [[Léon Foucault]], conceived as an experiment to demonstrate the [[Earth's rotation]]. If a long and heavy pendulum suspended from the high roof above a circular area is monitored over an extended period of time, its [[plane (geometry)|plane]] of [[oscillation]] appears to change spontaneously as the Earth makes its 24-hourly rotation. This effect is greatest at the poles and diminishes with lower latitude until it no longer exists at Earth's [[equator]].

The [[pendulum]] was introduced in 1851 and was the first experiment to give simple, direct evidence of the Earth's rotation. Foucault followed up in 1852 with a [[Foucault's gyroscope experiment|gyroscope experiment]] to further demonstrate the Earth's rotation. Foucault pendulums today are popular displays in [[science museum]]s and universities.<ref>{{cite journal|author=Oprea, John|title=Geometry and the Foucault Pendulum|journal=Amer. Math. Monthly|volume=102|issue=6|year=1995|pages=515–522|url=http://www.maa.org/programs/maa-awards/writing-awards/geometry-and-the-foucault-pendulum|doi=10.2307/2974765|url-status=live|archive-url=https://web.archive.org/web/20150402122208/http://www.maa.org/programs/maa-awards/writing-awards/geometry-and-the-foucault-pendulum|archive-date=2015-04-02|jstor=2974765|url-access=subscription}}</ref>

== History ==
[[File:Foucault Pendulum.jpg|thumb|left|upright|A print of the Foucault Pendulum, 1895]]
[[File:Foucault pendulum 1.webm|thumb|upright=1.2|Foucault Pendulum at [[COSI Columbus]] knocking over a ball]]

Foucault was inspired by observing a thin flexible rod on the axis of a [[lathe]], which vibrated in the same plane despite the rotation of the supporting frame of the lathe.<ref name=":0">{{Cite journal |last=Sommeria |first=Joël |date=2017-11-01 |title=Foucault and the rotation of the Earth |journal=Comptes Rendus Physique |series=Science in the making: The Comptes rendus de l’Académie des sciences throughout history |language=en |volume=18 |issue=9 |pages=520–525 |doi=10.1016/j.crhy.2017.11.003 |bibcode=2017CRPhy..18..520S |issn=1631-0705|doi-access=free }}</ref> 

The first public exhibition of a Foucault pendulum took place in February 1851 in the [[Paris meridian|Meridian]] of the [[Paris Observatory]]. A few weeks later, Foucault made his most famous pendulum when he suspended a {{convert|28|kg|lb|adj=on}} brass-coated lead [[bob (physics)|bob]] with a {{convert|67|m|ft|adj=mid|long}} wire from the dome of the [[Panthéon, Paris]]. 

Because the latitude of its location was <math>\phi = \mathrm{48^\circ 52' N}</math>, the plane of the pendulum's swing made a full circle in approximately <math display="inline">\frac{\mathrm{23h56'}}{\sin \phi} \approx \mathrm{31.8\,h} \;(\mathrm{31\,h\,50\,min})</math>, rotating clockwise approximately 11.3° per hour. The proper period of the pendulum was approximately <math display="inline">2\pi\sqrt{l/g}\approx 16.5 \,\mathrm{s}</math>, so with each oscillation, the pendulum rotates by about <math>9.05 \times 10^{-4} \mathrm{rad}</math>. Foucault reported observing 2.3 mm of deflection on the edge of a pendulum every oscillation, which is achieved if the pendulum swing angle is 2.1°.<ref name=":0" />

Foucault explained his results in an 1851 paper entitled ''Physical demonstration of the Earth's rotational movement by means of the pendulum'', published in the ''[[Comptes rendus de l'Académie des Sciences]]''. He wrote that, at the North Pole:<ref>{{Cite wikisource |wslanguage=fr |title=Démonstration physique du mouvement de rotation de la Terre au moyen du pendule |last=Foucault |first=Léon |year=1851}}</ref>
<blockquote>...an oscillatory movement of the pendulum mass follows an arc of a circle whose plane is well known, and to which the [[inertia]] 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.</blockquote>

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.<ref>{{cite web |url=http://www.parisenimages.fr/fr/galerie-collections/9232-13-pendule-foucault-du-pantheon-ceremonie-dinauguration-m-chaumie-ministre-linstruction-publique-brulant-fil-retenue-mettre-marche-pendule-1902 |title=The Pendulum of Foucault of the Panthéon. Ceremony of inauguration by M. Chaumié, minister of the state education, burnt the wire of balancing, to start the pendulum. 1902 |publisher=Paris en images |url-status=dead |archive-url=https://web.archive.org/web/20140821171755/http://www.parisenimages.fr/fr/galerie-collections/9232-13-pendule-foucault-du-pantheon-ceremonie-dinauguration-m-chaumie-ministre-linstruction-publique-brulant-fil-retenue-mettre-marche-pendule-1902 |archive-date=2014-08-21 }}</ref>

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]] before it reopened in 2000.<ref>{{cite web|last=Kissell|first=Joe|title=Foucault's Pendulum: Low-tech proof of Earth's rotation|url=http://itotd.com/articles/362/foucaults-pendulum/|publisher=Interesting thing of the day|access-date=March 21, 2012|date=November 8, 2004|url-status=live|archive-url=https://web.archive.org/web/20120312175754/http://itotd.com/articles/362/foucaults-pendulum/|archive-date=March 12, 2012}}</ref> 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.<ref>{{cite magazine |url=http://www.lexpress.fr/actualite/sciences/le-pendule-de-foucault-perd-la-boule_888228.html |title=Le pendule de Foucault perd la boule |first=Boris |last=Thiolay |magazine=L'Express |date=April 28, 2010 |language=fr |url-status=live |archive-url=https://web.archive.org/web/20100710123539/http://www.lexpress.fr/actualite/sciences/le-pendule-de-foucault-perd-la-boule_888228.html |archive-date=July 10, 2010 }}</ref><ref>{{cite news |last=Caulcutt |first=Clea |title=Foucault's pendulum is sent crashing to Earth |url=https://www.timeshighereducation.com/news/foucaults-pendulum-is-sent-crashing-to-earth/411529.article |access-date=August 10, 2024 |newspaper=[[Times Higher Education]] |date=13 May 2010 |url-access=limited |url-status=live |archive-url=https://web.archive.org/web/20240302021907/https://www.timeshighereducation.com/news/foucaults-pendulum-is-sent-crashing-to-earth/411529.article |archive-date=March 2, 2024}}</ref> 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.<ref>{{cite magazine|title=Foucault's Pendulum and the Paris Pantheon|url=https://www.atlasobscura.com/places/pantheon-paris|access-date=January 12, 2018|magazine=Atlas Obscura|url-status=live|archive-url=https://web.archive.org/web/20180112160124/https://www.atlasobscura.com/places/pantheon-paris|archive-date=January 12, 2018}}</ref>

== Mechanism ==
{{ multiple image
| image1   = Foucault-rotz.gif
| caption1 = Animation of a Foucault pendulum on the northern hemisphere, with the Earth's rotation rate and amplitude greatly exaggerated. The green trace shows the path of the pendulum bob over the ground (a rotating reference frame), while the bob moves in the corresponding vertical planes. The actual plane of swing appears to rotate relative to the Earth: sitting astride the bob like a swing, Coriolis fictitious force disappears: observer is in a "free rotational" reference. The wire should be as long as possible—lengths of {{convert|12|–|30|m|ft|-1|abbr=on}} are common.<ref>{{cite web|title=Foucault Pendulum|publisher=Smithsonian Encyclopedia|url=http://www.si.edu/Encyclopedia_SI/nmah/pendulum.htm|access-date=September 2, 2013}}</ref>
| image2   = Foucault pendulum animated.gif
| caption2 = Animated Foucault pendulum but with a trajectory on the ground which does not correspond to a bob launched without initial velocity
| image3   = Foucault pendulum at north pole animated.gif
| caption3 = A Foucault pendulum at the North Pole: The pendulum swings within a single plane as the Earth rotates beneath it
| image4   = Foucault pendulum plane of swing semi3D.gif
| caption4 = 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
}}

At either the [[Geographic North Pole]] or [[Geographic South Pole]], the plane of oscillation of a pendulum remains fixed relative to the [[Mach's principle|distant masses of the universe]]{{citation needed|date=January 2025}} 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 (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 plane of oscillation remains fixed relative to Earth. At other latitudes, the plane of oscillation [[Precession|precesses]] relative to Earth, but more slowly than at the pole; the angular speed, {{mvar|ω}} (measured in clockwise [[degree (angle)|degrees]] per sidereal day), is proportional to the [[sine]] of the [[latitude]], {{mvar|φ}}:

<math display="block">\omega=360^\circ\sin\varphi\ /\mathrm{day},</math>

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.<ref>{{cite web|title=Pendulum day|url=http://amsglossary.allenpress.com/glossary/search?id=pendulum-day1|url-status=dead|archive-url=https://web.archive.org/web/20070817221735/http://amsglossary.allenpress.com/glossary/search?id=pendulum-day1|archive-date=2007-08-17|website=Glossary of Meteorology| publisher=American Meteorological Society}}</ref><ref>{{Cite web|last1=Daliga|first1=K.|last2=Przyborski|first2=M. |last3=Szulwic|first3=J. |title=Foucault's Pendulum. Uncomplicated Tool In The Study Of Geodesy And Cartography |url=http://library.iated.org/view/DALIGA2015FOU|url-status=live|archive-url=https://web.archive.org/web/20160302024703/http://library.iated.org/view/DALIGA2015FOU|archive-date=2016-03-02|access-date=2015-11-02|website=library.iated.org}}</ref> 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&nbsp;mm each period.

A Foucault pendulum requires care to set up because imprecise construction can cause additional veering which masks the terrestrial effect. [[Heike Kamerlingh Onnes]] (Nobel laureate 1913) performed precise experiments and developed a fuller theory of the Foucault pendulum for his doctoral thesis (1879). He observed the pendulum to go over from linear to elliptic oscillation in an hour. By a [[Perturbation theory|perturbation analysis]], he showed that geometrical imperfection of the system or elasticity of the support wire may cause a beat between two horizontal modes of oscillation.<ref>{{cite journal |last1=Sommeria |first1=Joël |title=Foucault and the rotation of the Earth |journal=Comptes Rendus Physique |date=1 November 2017 |volume=18 |issue=9 |pages=520–525 |doi=10.1016/j.crhy.2017.11.003 |bibcode=2017CRPhy..18..520S |doi-access=free }}</ref> 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 [[:File:Le petit Parisien illustre 2nov1902.jpg|detail of the launch at the 50th anniversary in 1902]]).

Notably, veering of a pendulum was observed already in 1661 by [[Vincenzo Viviani]], a disciple of [[Galileo]], 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]] 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.<ref>{{Cite web | url=http://www.sas.org/E-Bulletin/2002-04-26/handsOnPhys/body.html | archive-url=https://web.archive.org/web/20090331105446/http://www.sas.org/E-Bulletin/2002-04-26/handsOnPhys/body.html | url-status=dead | archive-date=2009-03-31 |title = A Short, Driven, Foucault Pendulum}}</ref>

==Related physical systems==
[[File:Wheatstone Foucault device 256x256.png|thumb|The device described by Wheatstone.]]
Many physical systems precess in a similar manner to a Foucault pendulum. As early as 1836, the Scottish mathematician [[Edward Sang]] contrived and explained the precession of a spinning top.<ref>{{cite web | url=https://books.google.com/books?id=rtpQAAAAYAAJ&pg=PA105 | title=The Practical Mechanic's Journal | date=1857 }}</ref> In 1851, [[Charles Wheatstone]]<ref>[[Charles Wheatstone]] Wikisource: "[[s:Note relating to M. Foucault's new mechanical proof of the Rotation of the Earth|Note relating to M. Foucault's new mechanical proof of the Rotation of the Earth]]", pp. 65–68.</ref> described an apparatus that consists of a vibrating spring that is mounted on top of a disk so that it makes a fixed angle {{mvar|φ}} 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 {{mvar|φ}}.

Similarly, consider a nonspinning, perfectly balanced bicycle wheel mounted on a disk so that its axis of rotation makes an angle {{mvar|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 {{math|2π sin ''φ''}}.

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.<ref name="psb1">{{cite arXiv |last=Bharadhwaj |first=Praveen |eprint=1408.3047 |class=physics.pop-ph |title=Foucault precession manifested in a simple system |year=2014 }}</ref>

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 ''S''<sup>3</sup> in 4-dimensional [[Euclidean space]] with imaginary radius and imaginary timelike coordinate. Parallel transport of polarization vectors along such sphere gives rise to [[Thomas precession]], which is analogous to the rotation of the swing plane of Foucault pendulum due to parallel transport along a sphere ''S''<sup>2</sup> in 3-dimensional Euclidean space.<ref>{{cite journal | last1=Krivoruchenko | first1=M. I. | year=2009 | title=Rotation of the swing plane of Foucault's pendulum and Thomas spin precession: Two faces of one coin | journal=Phys. Usp. | volume=52 | issue=8| pages=821–829 |arxiv=0805.1136| bibcode=2009PhyU...52..821K| doi=10.3367/UFNe.0179.200908e.0873 | s2cid=118449576 }}</ref>

In physics, the evolution of such systems is determined by [[geometric phase]]s.<ref>"Geometric Phases in Physics", eds. [[Frank Wilczek]] and Alfred Shapere (World Scientific, Singapore, 1989).</ref><ref>L. Mangiarotti, G. [[Sardanashvily]], [https://books.google.com/books?id=-N6F44hlnhgC&dq=%22Berry+connection%22&pg=PA281 ''Gauge Mechanics''] (World Scientific, Singapore, 1998)</ref> Mathematically they are understood through parallel transport.

== Absolute reference frame for pendulum ==
{{main|Inertial frame of reference}}
The motion of a pendulum, such as the Foucault pendulum, is typically analyzed relative to an [[Inertial frame of reference]], approximated by the "fixed stars."<ref name="Pendulum">{{cite book |last1=Matthews |first1=Michael R. |last2=Gauld |first2=Colin F. |last3=Stinner |first3=Arthur |title=The Pendulum: Scientific, Historical, Philosophical and Educational Perspectives |year=2005 |publisher=Springer |isbn=978-1-4020-3525-8 |url=https://link.springer.com/book/10.1007/1-4020-3526-8 }}</ref>
These stars, owing to their immense distance from Earth, exhibit negligible motion relative to one another over short timescales, making them a practical benchmark for physical calculations. While fixed stars are sufficient for physical analyses, the concept of an absolute reference frame introduces philosophical and theoretical considerations.

'''Newtonian absolute space'''
* Isaac Newton proposed the existence of "absolute space," a universal, immovable reference frame independent of any material objects. In his ''Principia Mathematica'', Newton described absolute space as the backdrop against which true motion occurs.<ref name = "Sochi">{{cite web |last=Sochi |first=Taha |title=Absolute Frame in Physics |url=https://www.academia.edu/126444167/Absolute_Frame_in_Physics |website=Academia.edu |access-date=2025-01-04 }}</ref>
* This concept was criticized by later thinkers, such as Ernst Mach, who argued that motion should only be defined relative to other masses in the universe.<ref name="Sochi" />

'''[[Cosmic microwave background]] (CMB)'''
* The CMB, the remnant radiation from the Big Bang, provides a universal reference for cosmological observations. By measuring motion relative to the CMB, scientists can determine the velocity of celestial bodies, including Earth, relative to the universe's early state. This has led some to consider the CMB a modern analogue of an absolute reference frame.<ref name="Barbour">{{cite book |last=Barbour |first=Julian B. |title=Absolute or Relative Motion?: Volume 1, The Discovery of Dynamics: A Study from a Machian Point of View of the Discovery and the Structure of Dynamical Theories |year=1989 |publisher=Cambridge University Press |isbn=978-0521324670 }}</ref>

'''[[Mach's principle]] and distant masses'''
* [[Ernst Mach]] proposed that inertia arises from the interaction of an object with the distant masses in the universe. According to this view, the pendulum's frame of reference might be defined by the distribution of all matter in the cosmos, rather than an abstract absolute space.<ref name="Sochi" />
* The "distant masses of the universe" play a crucial role in defining the inertial frame, suggesting that the pendulum's apparent motion might be influenced by the collective gravitational effect of these masses. This perspective aligns with Mach’s principle, emphasizing the interconnectedness of local and cosmic phenomena.<ref name="Sochi" /><ref name="Barbour" />
* However, the connection between Mach's principle and Einstein's general relativity remains unresolved. Einstein initially hoped to incorporate Mach's ideas but later acknowledged difficulties in doing so.<ref name="MachInertia">{{cite web |last=Unknown |title=Spacetime Theories: Mach's Principle and Inertia |url=https://plato.stanford.edu/entries/spacetime-theories/#TwoInteMachIner |website=Stanford Encyclopedia of Philosophy |access-date=2025-01-04 }} One can see why the Machian interpretation Einstein hoped he could give to the curved spacetimes of his theory fails to be plausible, by considering a few simple ‘worlds’ permitted by GTR</ref>

'''[[General relativity]] and spacetime'''
* General relativity suggests that spacetime itself can serve as a reference frame. The pendulum’s motion might be understood as relative to the curvature of spacetime, which is influenced by nearby and distant masses. This view aligns with the concept of geodesics in curved spacetime.<ref name="Barbour" />
* The [[Lense-Thirring effect#Example: Foucault's pendulum|Lense-Thirring effect]],<ref name="Cartmell2020">{{cite journal |last=Cartmell |first=Matthew P. |last2=Smith |first2=James D. |title=Modelling and testing a laboratory-scale Foucault pendulum for relativistic frame-dragging measurements |journal=Proceedings of the Royal Society A |volume=476 |issue=2237 |year=2020 |pages=20200680 |doi=10.1098/rspa.2019.0680 |url=https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2019.0680 |access-date=2025-01-04 |pmc=7428043 }}</ref> a prediction of general relativity, implies that massive rotating objects like Earth can slightly "drag" spacetime,<ref name="Cartmell2024">{{cite journal |last=Cartmell |first=Matthew P. |last2=Smith |first2=James D. |title=The terrestrial measurement of relativistic frame-dragging with a Foucault pendulum |journal=Journal of Relativistic Physics |volume=48 |issue=2 |year=2024 |pages=123-145 |url=https://pureportal.strath.ac.uk/en/publications/the-terrestrial-measurement-of-relativistic-frame-dragging |access-date=2025-01-04 }}</ref> which could affect the pendulum’s oscillation. This effect, though theoretically significant, is currently too small to measure with a Foucault pendulum.

== Equation formulation for the Foucault pendulum ==
To model the Foucault pendulum, we consider a pendulum of length ''L'' and mass ''m'', oscillating with small amplitudes. In a reference frame rotating with Earth at angular velocity Ω, the Coriolis force must be included. The equations of motion in the horizontal plane (''x'', ''y'') are:

:<math>
\begin{aligned}
\ddot{x} + \omega_0^2 x &= 2\Omega \sin(\varphi) \dot{y}, \\
\ddot{y} + \omega_0^2 y &= -2\Omega \sin(\varphi) \dot{x},
\end{aligned}
</math>

where:
* <math>\omega_0 = \sqrt{\frac{g}{L}}</math> is the natural angular frequency of the pendulum,
* <math>\varphi</math> is the latitude,
* <math>g</math> is the acceleration due to gravity.

These coupled differential equations describe the pendulum's motion, incorporating the Coriolis effect due to Earth's rotation.<ref>{{cite web |title=Foucault Pendulum Details |url=https://www.phys.unsw.edu.au/~jw/pendulumdetails.html |publisher=UNSW Physics |access-date=2025-01-11}}</ref>

=== Precession rate calculation ===
The precession rate of the pendulum’s oscillation plane depends on latitude. The angular precession rate <math>\Omega_p</math> is given by:

:<math>\Omega_p = \Omega \sin(\varphi),</math>

where <math>\Omega</math> is Earth's angular rotation rate (approximately <math>7.2921 \times 10^{-5}</math> radians per second).<ref>{{cite web |title=Mathematical Derivations of the Foucault Pendulum |url=https://www.idc-online.com/technical_references/pdfs/mechanical_engineering/Mathematical_derivations_of_the_Foucault_pendulum.pdf |publisher=IDC Online |access-date=2025-01-11}}</ref>

=== Examples of precession periods ===

The time <math>T_p</math> for a full rotation of the pendulum’s plane is:

:<math>T_p = \frac{2\pi}{\Omega_p} = \frac{2\pi}{\Omega \sin(\varphi)}.</math>

Calculations for specific locations:

* '''Paris, France''' (latitude <math>\varphi \approx 48.8566^\circ</math>):
:<math>
\begin{aligned}
\Omega_p &= \Omega \sin(48.8566^\circ) \approx 7.2921 \times 10^{-5} \times 0.7547 \\
&\approx 5.506 \times 10^{-5} \, \text{radians/second}, \\
T_p &= \frac{2\pi}{5.506 \times 10^{-5}} \approx 114,105 \, \text{seconds} \\
&\approx 31.7 \, \text{hours}.
\end{aligned}
</math><ref>{{cite web |title=Foucault Pendulum Derivation |url=https://warwick.ac.uk/fac/sci/physics/intranet/pendulum/derivation/ |publisher=Warwick University |access-date=2025-01-11}}</ref>

* '''New York City, USA''' (latitude <math>\varphi \approx 40.7128^\circ</math>):
:<math>
\begin{aligned}
\Omega_p &= \Omega \sin(40.7128^\circ) \approx 7.2921 \times 10^{-5} \times 0.6523 \\
&\approx 4.757 \times 10^{-5} \, \text{radians/second}, \\
T_p &= \frac{2\pi}{4.757 \times 10^{-5}} \approx 132,000 \, \text{seconds} \\
&\approx 36.7 \, \text{hours}.
\end{aligned}
</math><ref>{{cite web |title=Foucault Pendulum Details |url=https://www.phys.unsw.edu.au/~jw/pendulumdetails.html |publisher=UNSW Physics |access-date=2025-01-11}}</ref>

These calculations show that the pendulum's precession period varies with latitude, completing a full rotation more quickly at higher latitudes.

== Installations ==
{{Further|List of Foucault pendulums}}

There are numerous Foucault pendulums at universities, science museums, and the like throughout the world. The [[United Nations General Assembly Building]] at the [[United Nations headquarters]] in New York City has one. The [[Oregon Convention Center]] pendulum is claimed to be the largest, its length approximately {{cvt|27|m}},<ref>{{cite web|url=http://jonesginzel.com/project/principia/|title=Kristin Jones - Andrew Ginzel|access-date=5 May 2018}}</ref><ref>{{cite web|url=http://ltwautomation.net/casestudies.html#Pendulum|title=LTW Automation Products|website=ltwautomation.net|access-date=5 May 2018|url-status=dead|archive-url=https://web.archive.org/web/20160429210824/http://ltwautomation.net/casestudies.html#Pendulum|archive-date=29 April 2016}}</ref> however, there are larger ones listed in the article, such as the one in Gamow Tower at the [[University of Colorado]] of {{cvt|39.3|m}}. There used to be much longer pendulums, such as the {{cvt|98|m}} pendulum in [[Saint Isaac's Cathedral]], [[Saint Petersburg]], [[Russia]].<ref>{{Cite web| url=https://visitmurmansk.info/en/the-first-foucault-pendulum-in-russia-beyond-the-arctic-circle/| title=The first Foucault pendulum in Russia, beyond the Arctic Circle| date=2018-06-14| access-date=2019-03-21| archive-date=2019-03-21| archive-url=https://web.archive.org/web/20190321085755/https://visitmurmansk.info/en/the-first-foucault-pendulum-in-russia-beyond-the-arctic-circle/| url-status=dead}}</ref><ref>[https://encyclopedia2.thefreedictionary.com/Foucault+pendulum Great Soviet Encyclopedia]</ref>

The experiment has also been carried out at the [[South Pole]], where it was assumed that the rotation of the Earth would have maximum effect.<ref>{{cite news |title=Here They Are, Science's 10 Most Beautiful Experiments |first=George |last=Johnson |url=https://www.nytimes.com/2002/09/24/science/here-they-are-science-s-10-most-beautiful-experiments.html?pagewanted=all&src=pm |newspaper=The New York Times |date=September 24, 2002 |access-date=September 20, 2012 |url-status=live |archive-url=https://web.archive.org/web/20120531225159/http://www.nytimes.com/2002/09/24/science/here-they-are-science-s-10-most-beautiful-experiments.html?pagewanted=all&src=pm |archive-date=May 31, 2012 }}</ref><ref>{{cite book |last=Baker |first=G. P. |year=2011 |title=Seven Tales of the Pendulum |pages=388 |publisher=[[Oxford University Press]] |isbn=978-0-19-958951-7}}</ref> 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 {{cvt|33|m}} and the bob weighed {{cvt|25|kg}}. 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 ==

* {{anl|Absolute rotation}}
* {{anl|Coriolis effect}}
* {{anl|Eötvös experiment}}
* {{anl|Geometric phase}}
* {{anl|Gyroscope}}
* {{anl|Inertial frame}}
* {{anl|Lariat chain}}
* {{anl|Precession}}

==References==
{{reflist|2}}

==Further reading==
*{{Cite book
 | last=Arnold
 | first=V.I.
 | title=Mathematical Methods of Classical Mechanics
 | publisher=Springer
 | year=1989
 | isbn=978-0-387-96890-2
 | page=[https://archive.org/details/mathematicalmeth0000arno/page/123 123]
 | url=https://archive.org/details/mathematicalmeth0000arno/page/123
 }}
*{{Cite book
 | last1=Marion
 | first1=Jerry B.
 | last2=Thornton
 | first2=Stephen T.
 | title=Classical dynamics of particles and systems
 | edition=4th
 | publisher=Brooks Cole
 | year=1995
 | isbn=978-0-03-097302-4
 | pages=[https://archive.org/details/classicaldynamic00mari_0/page/398 398–401]
 | url-access=registration
 | url=https://archive.org/details/classicaldynamic00mari_0/page/398
 }}
*{{Cite journal
 | last=Persson
 | first=Anders O.
 | title=The Coriolis Effect: Four centuries of conflict between common sense and mathematics, Part I: A history to 1885
 | journal=History of Meteorology
 | volume=2
 | year=2005
 | url=http://www.meteohistory.org/2005historyofmeteorology2/01persson.pdf
 | access-date=2006-04-27
 | archive-url=https://web.archive.org/web/20140411174448/http://www.meteohistory.org/2005historyofmeteorology2/01persson.pdf
 | archive-date=2014-04-11
 | url-status=dead
 }}

==External links==
{{Commons category|Foucault pendulums}}
* 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.
* "[http://www.animations.physics.unsw.edu.au/jw/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, [[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] {{Webarchive|url=http://arquivo.pt/wayback/20160525163952/http://www.calacademy.org/products/pendulum/index.html |date=2016-05-25 }} Foucault pendulum explanation, in friendly format
* [https://web.archive.org/web/20090202075110/http://electron.physics.buffalo.edu/ubexpo/Foucault%20Pendulum.html Foucault pendulum model] Exposition including a tabletop device that shows the Foucault effect in seconds.
* Foucault, M. L., [https://web.archive.org/web/20071120202153/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, retrieved 2007-10-31. Translation of his paper on Foucault pendulum.
* {{cite web |first1=William |last1=Tobin |title=The Life and Science of Léon Foucault |url=http://tobin.fr/foucault.html}}
* {{cite web |last=Bowley |first=Roger |title=Foucault's Pendulum |url=http://www.sixtysymbols.com/videos/foucault.htm |website=Sixty Symbols |publisher=[[Brady Haran]] for [[University of Nottingham]] |year=2010}}
* [http://www.labtrek.it/libroPF_rid.pdf Pendolo nel Salone] The Foucault Pendulum inside Palazzo della Ragione in Padova, Italy
* {{cite journal|journal = Am. J. Math. | first1=A. S. | last1=Chessin | title =On Foucault's Pendulum | year=1895 | volume=17|number=1|pages=81–88 | doi=10.2307/2369710 | jstor=2369710}} 
* {{cite journal | last=Bromwich | first=T. J. I'A. | title=On the Theory of Foucault's Pendulum, and of the Gyrostatic Pendulum | journal=Proceedings of the London Mathematical Society | volume=s2-13 | issue=1 | date=1914 | doi=10.1112/plms/s2-13.1.222 | pages=222–235 | url=http://doi.wiley.com/10.1112/plms/s2-13.1.222 | access-date=2025-05-30}}
* {{cite journal |journal=Am. J. Math. |first1=William Duncan |last1=MacMillan |s2cid=123717776 |title=On Foucault's Pendulum |volume=37 |issue=1 |year=1915 |pages=95–106 |doi=10.2307/2370259|jstor=2370259 }}
* {{cite journal|last1=Noble | first1= William J. |title= A direct treatment of the Foucault Pendulum | year= 1952 | pages=334–336|journal=Am. J. Phys.|doi=10.1119/1.1933230|volume=20| issue= 6 | bibcode= 1952AmJPh..20..334N }}
* {{cite journal|first1=E. O.|last1=Schulz-DuBois|title=Foucault Pendulum Experiment by Kamerlingh Onnes and degenerate perturbation theory|journal=Am. J. Phys.|volume=38|page=173|doi=10.1119/1.1976270|year=1970|issue=2 |bibcode=1970AmJPh..38..173S }}
* {{cite journal |last1=Somerville |first1=W. B. |title=The description of Foucault's pendulum |journal=Q. J. R. Astron. Soc. |year=1972 |volume=13 |pages=40–62 |bibcode=1972QJRAS..13...40S}}
*{{cite journal |title=Foucault Pendulum at the South Pole: Proposal For an Experiment to Detect the Earth's General Relativistic Gravitomagnetic Field |first1=Vladimir B. |last1=Braginsky |first2=Aleksander G. |last2=Polnarev |first3=Kip S. |last3=Thorne |journal=Phys. Rev. Lett. |volume=53 |issue=9 |page=863 |year=1984 |doi=10.1103/PhysRevLett.53.863 |bibcode=1984PhRvL..53..863B |url=https://authors.library.caltech.edu/3753/1/BRAprl84.pdf }}
* {{cite journal |first1=John B. |last1=Hard |first2=Raymond E. |last2=Miller |doi=10.1119/1.14972 |journal=Am. J. Phys. |volume=55 |issue=1 |year=1987 |page=67 |title=A simple geometric model for visualizing the motion of a Foucault pendulum|bibcode=1987AmJPh..55...67H}}
*{{cite journal |first1=H. Richard |last1=Crane |title=Foucault pendulum "wall clock" |journal=Am. J. Phys. |volume=63 |issue=1 |year=1995 |pages=33–39 |doi=10.1119/1.17765 |bibcode=1995AmJPh..63...33C}}
* {{cite journal |first1=U. |last1=Das |first2=B. |last2=Talukdar |first3=J. |last3=Shamanna |journal=Czechoslov. J. Phys. |volume=52 |issue=12 |year=2002 |pages=1321–1327 |title=Indirect Analytic Representation of Foucault's Pendulum |doi=10.1023/A:1021819627736 |bibcode=2002CzJPh..52.1321D|s2cid=118592240 }}
* {{cite journal |first1=Horacio R. |last1=Salva |first2=Rubén E. |last2=Benavides |first3=Julio C. |last3=Perez |first4=Diego J. |last4=Cuscueta |title=A Foucault's pendulum design |journal=Rev. Sci. Instrum. |volume=81 |issue=11 |year=2010 |pages=115102–115102–4 |doi=10.1063/1.3494611 |pmid=21133496 |bibcode=2010RScI...81k5102S }}
* {{cite journal|first1=M.|last1=de Icaza-Herrera| first2= V. M. |last2=Castano|title=Generalized Lagrangian of the parametric Foucault pendulum with dissipative forces | journal = Acta Mech.| volume=218 | pages=45–64|doi =10.1007/s00707-010-0392-8|year=2011|issue=1–2 }}
*{{cite book |last1=Daliga |first1=K. |last2=Przyborski |first2=M. |last3=Szulwic |first3=J. |chapter-url=https://www.researchgate.net/publication/277464397 |chapter=Foucault's Pendulum. Uncomplicated Tool in the Study of Geodesy and Cartography |isbn=978-84-606-8243-1 |year=2015 |title=EDULEARN15 Proceedings - 7th International Conference on Education and New Learning Technologies, Barcelona, Spain|publisher=IATED Academy }}
*{{cite journal|first1=Matthew P.|last1=Cartmell | first2=James E.|last2=Faller | first3=Nicholas A. |last3=Lockerbie |first4=Eva |last4=Handous | title =On the modelling and testing of a laboratory-scale Foucault pendulum as a precursor for the design of a high-performance measurement instrument|journal = Proc. R. Soc. A| volume =476|issue=2238|page=20190680| doi=10.1098/rspa.2019.0680 | year=2020|doi-access=free|pmid=32821234 |pmc=7428043|bibcode=2020RSPSA.47690680C }}

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[[Category:Pendulums]]
[[Category:Physics experiments]]
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