Spinning block begins to twist in mid-air?

[mrneglect]

Consider the following object:

The exact dimensions are unimportant, but its x, y and z dimensions must not be too similar (so I'm told).

I have been told that an object such as the one above will behave strangely when thrown up into the air whilst spinning.

If it is sent spinning on its x-axis then it will continue to do so until it falls back into your hands.
If it is sent spinning on its z-axis then it will continue to do so until it falls back into your hands.
But if it is sent spinning on its y-axis, it will begin to twist in mid-air.

At first I didn't believe it, and I certainly could see any physical reason for it to behave like this, but I've been trying this out by throwing my calculator up in the air and it works!

What's going on? Is it because the calculator isn't uniformly dense? Is it to do with air resistance? Is it to do with gravity? I'm told that it works in a vacuum with uniformly dense objects too, but I can't check that.

Please help to fix my intuition -- this one's baffling me!

 

[Doc Al]

The exact dimensions are unimportant, but its x, y and z dimensions must not be too similar (so I'm told).

The effect depends on the shape of the object. The rotational inertia about each principal axis must be different.

I have been told that an object such as the one above will behave strangely when thrown up into the air whilst spinning.

If it is sent spinning on its x-axis then it will continue to do so until it falls back into your hands.
If it is sent spinning on its z-axis then it will continue to do so until it falls back into your hands.
But if it is sent spinning on its y-axis, it will begin to twist in mid-air.

This is a consequence of Euler's equations for rotational motion. Motion about the axis with the intermediate rotational inertia--the y-axis in this case--is unstable.

In classical mechanics this is often called the "Tennis Racket Theorem".

Check this out: http://www.physics.usyd.edu.au/~cross/RacquetTheorem.mov"

 

[mrneglect]

Thank you, this is exactly what I was looking for!

 

[Nick89]

Cool, I never knew that either. I am always throwing things like that in the air, such as the remote, calculators, erasers, etc, and I've always wondered why it always does a twist, as well as a flip :p It's very hard to make it do a proper 'backflip' without the twist actually!

 

[Cantab Morgan]

Ever see the movie 2010 (sequel to 2001)? Roy Scheider and Jon Lithgow visit the long, thin Discovery spaceship tumbling end over end. It's rotating because the "hamster wheel" that Keir Dullea was running in in the original movie seized up. The thing is, the hamster wheel wasn't rotating along that axis.

However, as Doc Al pointed out, objects shaped like pencils don't like to rotate on the axis of their lead. They prefer to tumble end over end, because that rotation has a lower energy for the same momentum. It's kind of neat that they get the Physics right (or nearly right) in the movies sometimes.

 

[Bob S]

"The polhode rolls without slipping on the herpolhode..." from Goldstein "Classical Mechanics"

 

[rcgldr]

But if it is sent spinning on its y-axis, it will begin to twist in mid-air.

In classical mechanics this is often called the "Tennis Racket Theorem".

Tennis rackets and hammers can be tossed so they flip without twisting with the racket face or hammer head parallel to the axis of rotation, as long the the toss is done without imparting an initial twist. Either can also be tossed so it does a 1/2 twist or a full twist. Either can also be tossed and spun hard so it does a lot of twists. Divers, gymnasts, and trampolinists routinely do flips in the layout position without twisting.

 

[Nick89]

Of course you can stop it from twisting, but it's very hard. Even the slightest initial twist will send it spinning at least half a twist.

 

[rcgldr]

Of course you can stop it from twisting, but it's very hard. Even the slightest initial twist will send it spinning at least half a twist.

So an intial twist might be increased, but without the initial twist, a twist doesn't develop?

I'ts not always 1/2 twist per flip either. In this video, it's 2 1/2 flips, 1/2 twist mostly near the end. Driver (Peter Dunbreck) was OK.

http://www.youtube.com/watch?v=SFN_Gp1eHN0&fmt=18