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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?

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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).

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  • $\begingroup$ That's the physics, not the math behind it. You missed my question: "What's wrong with the math?" $\endgroup$
    – user541686
    Jul 26, 2012 at 6:23
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    $\begingroup$ Conservation of angular momentum is true, but your dθ/dt=0 equation only applies to a rigid body. $\endgroup$
    – bobuhito
    Jul 26, 2012 at 6:35
  • $\begingroup$ 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. $\endgroup$
    – user541686
    Jul 26, 2012 at 6:39
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    $\begingroup$ 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. $\endgroup$
    – bobuhito
    Jul 26, 2012 at 6:45
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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.

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  • $\begingroup$ but every part in the body of a cat has rotated by the same angle. $\endgroup$
    – Physiks lover
    Jul 26, 2012 at 14:21
  • $\begingroup$ @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. $\endgroup$
    – Mike Dunlavey
    Jul 26, 2012 at 15:18
  • $\begingroup$ do you honestly think cats have a spine where each half of the body can rotate n*360 degrees independently of the other? $\endgroup$
    – Physiks lover
    Jul 26, 2012 at 16:04
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    $\begingroup$ 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 $\endgroup$
    – Physiks lover
    Jul 26, 2012 at 20:29
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    $\begingroup$ 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. $\endgroup$
    – Ask About Monica
    Mar 5, 2013 at 21:14

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