# Table Top Physics - Tennis Racquets

Date: Mon, 30 Aug 2004 08:25:26 +1000
To: Martin Baker
From: ROD CROSS
Subject: Re: tennis racquets

Martin,

The tennis racquet theorem says that when you toss a tennis racquet
(or book or packet of cornflakes) in the air it will land in your
hand upside-down. That is, after a 360 degree rotation around one
axis, it does a 180 degree rotation about the long axis.

The mathematics is described by Euler's equations for a rotating
object in free space. All objects have 3 principle axes each with a
different moment of inertia in general, say I1 < I2 < I3. Rotation
about I1 and I3 is stable. Rotation about the I2 axis is unstable.

You can find the mathematical solutions in any book dealing with
dynamics of rigid objects. I like Deimel's book on mechanics of the
gyroscope, with polhode diagrams.

I have never seen a physics explanation of the tennis racquet
theorem. Somehow, the centripetal force along each axis generates a
torque about another axis which is either stabilising or
de-stabilising depending on the direction.

Rod Cross

From: Martin Baker
To: ROD CROSS
Subject: Re: tennis racquets
Date: Tue, 31 Aug 2004 13:56:42 +0100

Rod,

Thank you, I don't have a physics explanation either. My only guess would be since the axis of the combined rotation is about a diagonal which is not a symmetrical axis of the racket (not an eigenvector of its inertia tensor) Therefore I wonder if it is due to external forces such as air resistance rather than precession, since the air resistance is so much less on its narrow edge than its wide edge perhaps any small random variation will start it turning.

If its is alright I have put your message here, tennisRacquets Perhaps someone will send an explanation.

Thanks,

Martin

Date: Wed, 1 Sep 2004 15:16:28 +1000
To: Martin Baker
From: ROD CROSS
Subject: Re: tennis racquets

Martin,

It's not due to air resistance. The theory of it correctly says that a tennis racquet can be spun about 2 of the principal axes with only slight precessional wobbling, but the 3rd axis is unstable. Any slight asymmetry in throwing or tossing the racquet will cause the racquet to flip over - however, the rotation frequency increases about each of the other two stable axes (in a reference frame attached to the rotating racquet). When a racquet is half way through its flip it is rotating edge-on.

When tossing a book, it is relatively easy to toss it with 2 hands in such a way that it doesn't flip over since the initial asymmetry can be reduced almost to zero. A one-handed toss tends to introduce greater asymmetry, then the book will flip. I think it is a really neat solution of Euler's equations. It is something that most tennis
players know about but as a physicist and a tennis player I'm embarrased that I can't give a simple physics explanation. It's probably too complicated, like most things involving precession and other spinning objects.

Rod

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

3d Games: Physics -

Game Physics - This book has some useful stuff, its more of a textbook, not a step by step guide (although it does have a disc with a lot of C++ code). About the first third of the book is a physics textbook with theoretical exercises, the middle bit covers physics engine topics, and the last third of the book covers mathematical topics. I think I would use this book as a reference book to lookup the theory behind something I might be working on rather than a book to work through in order.

Commercial Software Shop

Where I can, I have put links to Amazon for commercial software, not directly related to the software project, but related to the subject being discussed, click on the appropriate country flag to get more details of the software or to buy it from them.

 Dark Basic Professional Edition - It is better to get this professional edition This is a version of basic designed for building games, for example to rotate a cube you might do the following: make object cube 1,100 for x=1 to 360 rotate object 1,x,x,0 next x Game Programming with Darkbasic - book for above software

This site may have errors. Don't use for critical systems.