On the spinning motion of the hovering magnetic top

Abstract

In this paper we analyze the spinning motion of the hovering magnetic top. We have observed that its motion looks different from that of a classical top. A classical top rotates about its own axis which precesses around a vertical fixed external axis. The hovering magnetic top, on the other hand, has its axis slightly tilted and moves rigidly * Also with the Department of Physics, University of California, San Diego, La Jolla, 92093 CA, USA 1 as a whole about the vertical axis. We call this motion synchronous, because in a stroboscopic experiment we see that a point at the rim of the top moves synchronously with the top axis.We show that the synchronous motion may be attributed to a small deviation of the magnetic moment from the symmetry axis of the top. We show that as a consequence, the minimum angular velocity required for stability is given by 4µHI 1 /I 2 3 for I 3 > I 1 and by µH/(I 3 − I 1 ) for I 3 < I 1 . Here, I 3 and I 1 are the principal and secondary moments of inertia, µ is the magnetic moment, and H is the magnetic field. For comparison, the minimum angular for a classical top is given by 4µHI 1 /I 2 3 both for I 3 < I 1 and for I 3 > I 1 . We also give experimental results that were taken with a top whose moment of inertia I 1 can be changed. These results show very good agreement with our calculations.

Dynamics of a Levitron under a periodic magnetic forcing

“…After a while, precession and rotation start to synchronize and, almost at the limit of stability, when the velocity has slowed down and the angle of precession has increased, the precession and spin rates become almost the same. This coupling between precession and spinning was observed by Flanders et al 7 and was explained by considering a small angle D between the magnetic moment and the top axis. The synchronization of precession and rotation is clearly noticeable in our video recordings [an example (0A_90s.mov) is included in the online supplement to this article 10 ].…”

“…The synchronous precession in the absence of an external field was already observed and explained by Flanders et al 7 as due to the small misalignment between the axis of the top and its magnetic dipole moment. Figure 4(a) shows a schematic view of the relative directions of these vectors.…”

“…As we have already mentioned, Flanders et al 7 showed that a small deviation of the magnetic moment from the symmetry axis of the top synchronizes the precession and the rotation of the top, in a way similar to the motion of the moon around the earth. In order to test this prediction, we ran several simulations for different values of D, the angle between the magnetic moment and the top axis.…”

“…2, [4][5][6][7][8] Once the top is in a stable state of levitation, air friction eventually decreases the spin rate. After about 2 min, the angular velocity drops below the lower stability limit and the levitation ends.…”

“…This deviation was already proposed by Flanders et al as an explanation of the fact that the top precesses synchronously with the spin of the top around itself. 7 This paper is organized as follows. In Secs.…”