Tennis racket theorem: Difference between revisions

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[[File:tennis_racket_theorem.gif|thumb|upright=1.5|link={{filepath:tennis_racket_theorem.ogv}}|Composite video of a tennis racquet rotated around the three axes – the intermediate one flips from the light edge to the dark edge]]

[[File:Théorie Nouvelle de la Rotation des Corps.jpg|thumb|Title page of "Théorie Nouvelle de la Rotation des Corps", 1852 printing]]

The '''tennis racket theorem''' or '''intermediate axis theorem''', is a kinetic phenomenon of [[classical mechanics]] which describes the movement of a [[rigid body]] with three distinct [[principal moments of inertia]]. It has also been dubbed the '''Dzhanibekov effect''', after [[Soviet Union|Soviet]] [[cosmonaut]] [[Vladimir Dzhanibekov]], who noticed one of the theorem's [[logical consequence]]s whilst in space in 1985.<ref>[http://oko-planet.su/science/sciencehypothesis/15090-yeffekt-dzhanibekova-gajka-dzhanibekova.html Эффект Джанибекова (гайка Джанибекова)], 23 July 2009 {{in lang|ru}}. The software can be downloaded [http://live.cnews.ru/forum/index.php?s=5091d296ac0d22ad6b6e9712f3b0edbe&act=Attach&type=post&id=87112 from here]</ref> The effect was known for at least 150 years prior, having been described by [[Louis Poinsot]] in 1834<ref>Poinsot (1834) [https://archive.org/details/thorienouvelled00poingoog/page/n9 ''Theorie Nouvelle de la Rotation des Corps''], Bachelier, Paris</ref><ref>{{cite AV media |publisher = Veritasium | title = The Bizarre Behavior of Rotating Bodies, Explained | date = September 19, 2019 | url = https://www.youtube.com/watch?v=1VPfZ_XzisU | access-date = February 16, 2020 | people = [[Derek Muller]] }}</ref> and included in standard physics textbooks such as [[Classical Mechanics (Goldstein)|''Classical Mechanics'']] by [[Herbert Goldstein]] throughout the 20th century.

The theorem describes the following effect: rotation of an object around its first and third [[Moment of inertia#Principal axes|principal axes]] is stable, whereas rotation around its second principal axis (or intermediate axis) is not.

This can be demonstrated by the following experiment: holdHold a tennis racket at its handle, with its face being horizontal, and throw it in the air such that it performs a full rotation around its horizontal axis perpendicular to the handle (ê₂ in the diagram), and then catch the handle. In almost all cases, during that rotation the face will also have completed a half rotation, so that the other face is now up. By contrast, it is easy to throw the racket so that it will rotate around the handle axis (ê₁) without accompanying half-rotation around another axis; it is also possible to make it rotate around the vertical axis perpendicular to the handle (ê₃) without any accompanying half-rotation.

The experiment can be performed with any object that has three different moments of inertia, for instance with a (rectangular) book, remote control, or smartphone. The effect occurs whenever the [[axis of rotation]] differs only slightly from the object's second principal axis; air resistance or gravity are not necessary.<ref>{{Cite book |url={{google books |plainurl=yes |id=uVSYswEACAAJ |page=151}} |title=Classical Mechanics with Calculus of Variations and Optimal Control: An Intuitive Introduction |last=Levi |first=Mark |publisher=American Mathematical Society |year=2014 |isbn=9781470414443 |pages=151–152 }}</ref>

== Theory ==