Spinning top undergoing nutation

[techmologist]

Suppose you have a symmetric top precessing and nutating under the torque due to gravity. My question is this: does the spin rate of the top (the rate at which it spins on its axis of symmetry) vary with time or is it constant? I am assuming the 'foot' of the top is fixed and that the motion about this fixed point is frictionless.

 

[Meir Achuz]

As the top nutates, The spin rate changes periodically to conserve energy.

 

[techmologist]

As the top nutates, The spin rate changes periodically to conserve energy.

Thanks for the reply, clem. I have been trying to understand the motion of a heavy top by freshman-level methods, namely, requiring that all the torques on individual mass elements of the top add up to the torque of gravity. This technique works well for the case of torque-free precession, where you can guess what the motion looks like. In that case the spin rate is constant, and I just assumed it was for a heavy top, too. I had also assumed the faster precession rate at the low point in the nutation cycle was enough gain in kinetic energy to balance the loss of potential energy. For some reason, I can't help but think of the spin rate as a kind of "intrinsic" property of the top. After all, it was determined when you twisted it with your thumb and forefinger, or pulled the string in the case of a gyroscope. How does the top "know" to speed up and slow down its spinning as it bobs up and down? I know that's a silly question, but I still find this periodic spin rate kinda spooky. Do you know of any book or website that gives an explanation of this beyond just solving the equations of motion?