Tennis racket theorem: Difference between revisions

Revision as of 14:32, 12 September 2022

Revision as of 14:03, 1 October 2022

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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 result in [[classical mechanics]] describing the movement of a [[rigid body]] with three distinct [[principal moments of inertia]]. It is also dubbed the '''Dzhanibekov effect''', after [[Soviet Union|Soviet]] [[cosmonaut]] [[Vladimir Dzhanibekov]] who noticed one of the theorem's [[logical consequence]]s while 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> although the effect was already known for at least 150 years before that, and was included in a book 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>

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

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This can be demonstrated with the following experiment: hold a tennis racket at its handle, with its face being horizontal, and try to throw it in the air so that it will perform a full rotation around the horizontal axis perpendicular to the handle, and try to 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 in the diagram) 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 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 ==