Animations by Iván Tello and Isaac Frame. Special thanks to: Astronaut Don Pettit, Henry Reich of MinutePhysics, Grant Sanderson of 3blue1brown, Vert Dider (Russian YouTube channel). Part of this video was sponsored by LastPass (ve42.co/LP).
Bizarre behavior of rotating bodies explained
Spinning objects have strange instabilities known collectively as the Dzhanibekov Effect or the "Tennis Racket Theorem." This investigation offers an intuitive explanation.
- The Twisting Tennis Racket: Ashbaugh, M.S., Chicone, C.C. & Cushman, R.H. J Dyn Diff Equat (1991) 3: 67. doi.org/10.1007/BF01049489
- Janibekov’s effect and the laws of mechanics: Petrov, A.G. & Volodin, S.E. Dokl. Phys. (2013) 58: 349. https://doi.org/10.1134/S102833581308...
- Tumbling Asteroids: Prave et al. https://doi.org/10.1016/j.icarus.2004...
- The Exact Computation of the Free Rigid Body Motion and Its Use in Splitting Methods: SIAM J. Sci. Comput., 30(4), 2084–2112: E. Celledoni, F. Fassò, N. Säfström, and A. Zanna: doi.org/10.1137/070704393
Discussion
This is a further discussion by Henry Reich that helps summarize why Axes 1 and 3 are generally stable while Axis 2 is not:
In general, imagine that, because the object can rotate in a bunch of different directions, the components of energy and momentum could be free to change, while the total momentum stays constant.
This is a further discussion by Henry Reich that helps summarize why Axes 1 and 3 are generally stable while Axis 2 is not:
In general, imagine that, because the object can rotate in a bunch of different directions, the components of energy and momentum could be free to change, while the total momentum stays constant.
However, in the case of Axis 1, the kinetic energy is the highest possible for a given angular momentum. And in the case of Axis 3, the kinetic energy is the lowest possible for a given angular momentum.
- (This can easily be shown from conservation of energy and momentum equations. And this is also fairly intuitive from the fact that kinetic energy is proportional to velocity squared, while momentum is proportional to velocity. So in the case of Axis 1, the smaller masses will have to be spinning faster for a given momentum. They will thus have more energy, and vice versa for Axis 3, where all the masses are spinning: The energy will be lowest).
In fact, this is a strict inequality: If the energy is highest possible, there are no other possible combinations of momenta other than L2=L3=0, and vice versa for [L1] if the energy is the lowest possible.
Because of this, in the case of Axis 1, the energy is so high that there simply aren't any other possible combinations of angular momentum components L1, L2, and L3. The object would have to lose energy in order to spin differently. And in the case of Axis 3, the energy is so low that there likewise is no way for the object to be rotating other than purely around Axis 3. It would have to gain energy.
However, there's no such constraint for Axis 2 since the energy is somewhere in between the min and max possible. This, together with the centrifugal effects, means that the components of momentum do change.
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