The Mysterious Stabilization of a Spinning Top

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Everyone has seen a spinning top in action, and maybe some of you have also seen how they're able to stabilize completely like shown in the attached picture. I do get that they're hard to stop spinning because of conservation of angular momentum, but that they should be able to stabilize just seems weird.
 

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I think you misunderstood me. I'm asking how it's able to get back into the position where it stands completely upright after it's been struck and started to preces at angle. I am pretty sure I have seen that happen.
 
My immediate guess is that it might be useful to consider the consequences of the contact point not being a perfect mathematical point, and the resulting interaction between the angular momenta of the top and the surface on which it spins...
 

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