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To explain why a falling cat can turn by 180 degree without external torque and without violation of the conservation of angular momentum, one usually models the cat as two cylinders as in

http://en.wikipedia.org/wiki/Falling_cat_problem

This may explain the turn. However I often heard contrary to this that she can rotate her body simply because she rotates her tail very fast into the opposite direction (and essentially keeps the rest of the body rigid).

So, what effect does the tail have in reality? Is there any detailed model, which takes the tail rotation into account and calculates how large its effect is?

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The two answers are physically equivalent. If you watch the Wikipedia animation

enter image description here

you see that the fastest part of her body on the top animation is the tail. It makes a substantial contribution to the angular momentum. At any rate, the angular momentum of the tail is included in the angular momentum of the "back cylinder". The trick the cat needs to achieve the task is to perform a fast relative motion of the back of her body (including, importantly, the tail) and the front of her body.

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The video on video.nationalgeographic.com/video/animals/mammals-animals/cats/… shows that the tail is actively moved relative to the body during the rotation (in at least one cat :-), so the situation is a bit more complicated than the animation. I would guess that because the weight of the tail is small compared to the body it is just used for fine control. –  John Rennie Apr 30 '12 at 16:14
    
Sorry, John, but I don't follow your logic. The relative motion of the back of her body (especially the tail) with respect to the rest of the body (the front) is the only thing that the cat may be doing by controlling her muscles and it is also what this animation shows. All other aspects of the motion are dictated by the changing total momentum and conserved angular momentum. In what sense can the "situation" be more complicated than the animation? The precise motion may have a bit different shape but the animation captures 100% of the physics related to the change of the orientation. –  Luboš Motl Apr 30 '12 at 16:21
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If the tail were heavy enough the cat could hold it's body/legs rigid and just rotate its tail. Clearly this is not the case and the cat must flex its spine. I think the OP is asking how much rotation could be achieved by motion at the tail/body joint. –  John Rennie Apr 30 '12 at 16:30
    
OK but there isn't any universal number for all cats how independently they may turn their tail relatively to the spine, or how large a percentage of the change of the angular momentum is guaranteed by the tail. At least this isn't a question of physics because one would have to define what she exactly means by a cat. The tail is a non-negligible part of the effect because while it's light, its center-of-mass is further than the bulk of the body from the relevant horizontal axis. At any rate, if one knows the exact geometry and masses, she may use the Wikipedia model to calculate the numbers. –  Luboš Motl Apr 30 '12 at 18:24
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