Tennis racket theorem: Difference between revisions

Revision as of 11:25, 14 November 2024

Revision as of 12:05, 14 November 2024

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The tennis racket effect occurs when <math>\omega(0)</math> is very close to a saddle point. The body would linger near the saddle point, then rapidly move to the other saddle point, near <math>\omega(T/2)</math>, linger again for a long time, and so on. The motion repeats with period <math>T</math>.

The above analysis is all done in the perspective of an observer which is rotating with the body. An observer watching the body's motion in free space would see its angular momentum vector <math>\vec L = I\vec \omega</math> conserved, while both its angular velocity vector <math>\vec \omega(t)</math> and its moment of inertia <math>I(t)</math> undergoingundergo complicated motions in space. At the beginning, the observer would see both <math>\vec \omega(0), \vec L</math> mostly aligned with the second major axis of <math>I(0)</math>. After a while, the body performs a complicated motion and ends up with <math>I(T/2), \vec \omega(T/2)</math>, and again both <math>\vec L, \vec \omega(T/2)</math> are mostly aligned with the second major axis of <math>I(T/2)</math>.

Consequently, there are two possibilities: either the rigid body's second major axis is in the same direction, or it has reversed direction. If it is still in the same direction, then <math>\vec\omega(0), \vec\omega(T/2)</math> viewed in the rigid body's reference frame are also mostly in the same direction. However, we have just seen that <math>\omega(0)</math> and <math>\omega(T/2)</math> are near opposite saddle points <math>(0, \pm L/I_2, 0)</math>. Contradiction.