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	~~{{expert-subject\|1=physics\|date=April 2012}}~~		Issac is the identify his mothers and fathers gave him but persons normally misspell it. A single of his favored hobbies is looking into [http://mondediplo.com/spip.php?page=recherche&recherche=cryptography cryptography] but he does not have the time lately. He employed to be unemployed but now he is an business office clerk. Delaware has constantly been his residing spot. Check out out the most recent information on his website: http://www.granvall.com/asicsbaratas/asics-gel-padel-exclusive-2012-018347927.aspx<br><br>Take a look at my web site - [http://www.granvall.com/asicsbaratas/asics-gel-padel-exclusive-2012-018347927.aspx asics gel padel exclusive 2012]
	~~The '''tennis racket theorem'''~~ is a result in [[classical mechanics]] describing movement of a rigid body with three distinct [[Angular momentum\|angular momenta]]. It also dubbed '''Dzhanibekov effect''' named after [[Russian people\|Russian]] [[astronaut]] [[Vladimir Dzhanibekov]] who discovered the ~~theorem's consequences while in space in 1985~~.~~<ref>~~[http://~~oko-planet~~.su/~~science/sciencehypothesis/15090-yeffekt-dzhanibekova-gajka-dzhanibekova~~.~~html]</ref>~~

	~~== Qualitative Proof~~ ==
	~~The tennis racket theorem can be qualitatively analysed with the help of [[Euler's equations (rigid body dynamics)\|Euler's equations~~]].

	~~Under torque free conditions, they take~~ the ~~following form:~~
	~~:<math>~~
	~~\begin{align}~~
	~~I_1\dot{\omega}_{1}&=(I_2-I_3)\omega_2\omega_3~~~~~~~~~~~~~~~~~~~~\text{(1)}\\~~
	~~I_2\dot{\omega}_{2}&=(I_3-I_1)\omega_3\omega_1~~~~~~~~~~~~~~~~~~~~\text{(2)}\\~~
	~~I_3\dot{\omega}_{3}&=(I_1-I_2)\omega_1\omega_2~~~~~~~~~~~~~~~~~~~~\text{(3)}~~
	~~\end{align}~~
	~~</math>~~


	~~Let <math> I_1 > I_2 > I_3 </math>~~

	~~Consider the situation when the object is rotating about axis with moment of inertia <math>I_1</math>~~. ~~To determine the nature of equilibrium, assume small initial angular velocities along the other two axes. As a result, according~~ to ~~equation (1), <math>~\dot{\omega}_{1}</math>~~ is ~~very small~~. ~~Therefore~~ the ~~time dependence of <math>~\omega_1</math> may be neglected.~~

	~~Now, differentiating equation (2) and substituting <math>\dot{\omega}_3</math> from equation (3),~~
	:~~<math>~~
	~~\begin{align}~~
	~~I_2 I_3 \ddot{\omega}_{2}&= (I_3-I_1) (I_1-I_2) \omega_1\omega_{2}\\~~
	~~\text{i.e.}~~~~ \ddot{\omega}_2 &= \text{(negative quantity)} \times \omega_2~~
	~~\end{align}~~
	</~~math>~~

	~~Note that <math>\omega_2<~~/~~math> is being opposed and so rotation around this axis is stable for the object~~.

	~~Similar reasoning also gives that rotation around axis with moment of inertia <math>I_3</math> is also stable~~.

	~~Now apply the same thing to axis with moment of inertia <math>I_2<~~/~~math>. This time <math>\dot{\omega}_{2}<~~/~~math> is very small. Therefore the time dependence of <math>~\omega_2</math> may be neglected.~~

	~~Now, differentiating equation (1) and substituting <math>\dot{\omega}_3</math> from equation (3),~~
	~~:<math>~~
	~~\begin{align}~~
	~~I_1 I_3 \ddot{\omega}_{1}&= (I_2~~-~~I_3) (I_1~~-~~I_2) \omega_1\omega_{2}\\~~
	~~\text{i.e~~.~~}~~~~ \ddot{\omega}_1 &= \text{(positive quantity)} \times \omega_1~~
	~~\end{align}~~
	<~~/math~~>

	~~Note that~~ <~~math~~>~~\omega_1</math> is '''not''' opposed and so rotation around this axis is '''unstable'''. Therefore even~~ a ~~small disturbance along other axes causes the object to 'flip'.~~

	~~== See also ==~~
	*[[Euler angles]]
	*[[Moment of inertia]]
	*[[Poinsot's ellipsoid]]
	*[[Polhode]]

	~~== References ==~~
	~~{{reflist}}~~
	* Mark S. Ashbaugh, Carmen C. Chicone and Richard H. Cushman, [http://math.ucalgary.ca/files/publications/cushman/tennis.pdf The Twisting Tennis Racket], ''Journal of Dynamics and Differential Equations'', Volume 3, Number 1, 67-~~85 (1991).~~
	* [http://www.~~youtube~~.com/~~watch?v=L2o9eBl_Gzw Dzhanibekov effect video] demonstrated on the [[International Space Station]]~~



	~~{{classicalmechanics~~-~~stub}}~~
	~~[[Category:Classical mechanics]]~~
	~~[[Category:Physics theorems]~~]

en>David Eppstein: stub sort

2012-08-29T07:14:44Z

stub sort

New page

{{expert-subject|1=physics|date=April 2012}}
The '''tennis racket theorem''' is a result in [[classical mechanics]] describing movement of a rigid body with three distinct [[Angular momentum|angular momenta]]. It also dubbed '''Dzhanibekov effect''' named after [[Russian people|Russian]] [[astronaut]] [[Vladimir Dzhanibekov]] who discovered the theorem's consequences while in space in 1985.<ref>[http://oko-planet.su/science/sciencehypothesis/15090-yeffekt-dzhanibekova-gajka-dzhanibekova.html]</ref>

== Qualitative Proof ==
The tennis racket theorem can be qualitatively analysed with the help of [[Euler's equations (rigid body dynamics)|Euler's equations]].

Under torque free conditions, they take the following form:
:<math>
\begin{align}
I_1\dot{\omega}_{1}&=(I_2-I_3)\omega_2\omega_3~~~~~~~~~~~~~~~~~~~~\text{(1)}\\
I_2\dot{\omega}_{2}&=(I_3-I_1)\omega_3\omega_1~~~~~~~~~~~~~~~~~~~~\text{(2)}\\
I_3\dot{\omega}_{3}&=(I_1-I_2)\omega_1\omega_2~~~~~~~~~~~~~~~~~~~~\text{(3)}
\end{align}
</math>

Let <math> I_1 > I_2 > I_3 </math>

Consider the situation when the object is rotating about axis with moment of inertia <math>I_1</math>. To determine the nature of equilibrium, assume small initial angular velocities along the other two axes. As a result, according to equation (1), <math>~\dot{\omega}_{1}</math> is very small. Therefore the time dependence of <math>~\omega_1</math> may be neglected.

Now, differentiating equation (2) and substituting <math>\dot{\omega}_3</math> from equation (3),
:<math>
\begin{align}
I_2 I_3 \ddot{\omega}_{2}&= (I_3-I_1) (I_1-I_2) \omega_1\omega_{2}\\
\text{i.e.}~~~~ \ddot{\omega}_2 &= \text{(negative quantity)} \times \omega_2
\end{align}
</math>

Note that <math>\omega_2</math> is being opposed and so rotation around this axis is stable for the object.

Similar reasoning also gives that rotation around axis with moment of inertia <math>I_3</math> is also stable.

Now apply the same thing to axis with moment of inertia <math>I_2</math>. This time <math>\dot{\omega}_{2}</math> is very small. Therefore the time dependence of <math>~\omega_2</math> may be neglected.

Now, differentiating equation (1) and substituting <math>\dot{\omega}_3</math> from equation (3),
:<math>
\begin{align}
I_1 I_3 \ddot{\omega}_{1}&= (I_2-I_3) (I_1-I_2) \omega_1\omega_{2}\\
\text{i.e.}~~~~ \ddot{\omega}_1 &= \text{(positive quantity)} \times \omega_1
\end{align}
</math>

Note that <math>\omega_1</math> is '''not''' opposed and so rotation around this axis is '''unstable'''. Therefore even a small disturbance along other axes causes the object to 'flip'.

== See also ==
*[[Euler angles]]
*[[Moment of inertia]]
*[[Poinsot's ellipsoid]]
*[[Polhode]]

== References ==
{{reflist}}
* Mark S. Ashbaugh, Carmen C. Chicone and Richard H. Cushman, [http://math.ucalgary.ca/files/publications/cushman/tennis.pdf The Twisting Tennis Racket], ''Journal of Dynamics and Differential Equations'', Volume 3, Number 1, 67-85 (1991).
* [http://www.youtube.com/watch?v=L2o9eBl_Gzw Dzhanibekov effect video] demonstrated on the [[International Space Station]]

{{classicalmechanics-stub}}
[[Category:Classical mechanics]]
[[Category:Physics theorems]]

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en>David Eppstein: stub sort