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