Editing Foucault pendulum (section)

== 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>