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Let θ be the orientation (angle) of a body (such as a cat), and let ω be its angular velocity.

It is well-known that θ can change even when the body is not rotating, using the conservation of angular momentum; that is, even when ω = dθ/dt = 0. That's how cats land on their feet so well.

But how can θ possibly ever change, when its derivative is zero?! What's wrong with the math?

user541686
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2 Answers2

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A rigid body can't change it's angle, but a cat is not rigid (it can move one part in one direction and other parts in the opposite direction, and effectively wiggle around the full circle).

bobuhito
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  • That's the physics, not the math behind it. You missed my question: "What's wrong with the math?" – user541686 Jul 26 '12 at 06:23
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    Conservation of angular momentum is true, but your dθ/dt=0 equation only applies to a rigid body. – bobuhito Jul 26 '12 at 06:35
  • Ooh... wow, that just blew my mind. So angular velocity isn't defined (or at least not the way I expect) for non-rigid bodies? +1 that explains it, thanks. – user541686 Jul 26 '12 at 06:39
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    The simplest non-rigid body would effectively need two theta angles (using some pivot point). The angular momentum conservation law would then relate those two angles and be much more complicated. – bobuhito Jul 26 '12 at 06:45
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Good answer from bobuhito. Here's another explanation. Satellites have reaction wheels (which are not gyroscopes) to help them change orientation.

If you sit still on a rotating stool, and you want to change direction, and you are holding a long heavy rod, simply hold the rod over your head and rotate it horizontally a couple times.

Your total angular momentum at all times is zero, but that's because there's a positive angular momentum in the rod, balanced by a negative one in your body. When you stop turning it, both you and the rod have changed direction.

Mike Dunlavey
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  • but every part in the body of a cat has rotated by the same angle. – Physiks lover Jul 26 '12 at 14:21
  • @Physikslover: But simplify the cat down to two rotating masses, where one mass is larger than the other. One has rotated one way, one has rotated the other. When they come into alignment again, they are no longer pointing in the original direction. If you are sitting on a stool with a weight in one hand, simply swing the weight around your head a few times - same thing. – Mike Dunlavey Jul 26 '12 at 15:18
  • do you honestly think cats have a spine where each half of the body can rotate n*360 degrees independently of the other? – Physiks lover Jul 26 '12 at 16:04
  • @Physikslover: Relative rotation does not require a swivel joint. Have you ever tried to swing a hula-hoop around your waist? It's the same motion. If you do that while standing on a frictionless platform, you will turn, and when you stop, you will be facing in a new direction. Alternatively, sit on the bar stool and swing your feet in a circle. – Mike Dunlavey Jul 26 '12 at 16:11
  • sure, likewise for the cat's body you can have one half rotating in one direction, the other in the opposite. But they realign so there is no angular displacement between them, and that's my point. I can sit on a bar stool and twist my body in one direction, the bar stool going in the other. But when I realign my body with the bar stool, I end up pointing in the same direction in the room as I started out with. – Physiks lover Jul 26 '12 at 16:28
  • @Physikslover: No need to reason about it, try it. Sit on a bar stool. Hold a heavy weight in one hand, hanging at your side. Then swing it up to horizontally in front of you (call that 12 oclock). Then swing it horizontally to the 3:00 position. That will of course turn you somewhat to the left. Then drop the weight back down to your side, where it started from. Now you are just as you were when you started, except you remain oriented to the left of where you started. – Mike Dunlavey Jul 26 '12 at 16:56
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    that only works because of friction which is an external torque and so the momentum of me and the bar stool isn't conserved. Your thought experiment wouldn't work on a frictionless bar stool – Physiks lover Jul 26 '12 at 20:29
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    It would. The 90 degree arc when it's fully extended contributes far more angular momentum than the 90 degree arc backwards does when it's near your center of gravity. – Ask About Monica Mar 05 '13 at 21:14