Martin D. Simon, UCLA Department of Physics,
msimon@physics.ucla.edu,
Lee O. Heflinger, Torrance CA, and S. L. Ridgway, Santa Monica CA
Published in the American Journal of Physics, April 1997
copyright AAPT 1997
The stability of the Levitron cannot be explained if the top's axis has a fixed direction in space. Stability against flipping is not enough. Gyroscopic precession around the local magnetic field direction is necessary. An analysis and numerical integration of the equations of motion for an experimental stemless top that includes gyroscopic precession around the local magnetic field lines predict that the top will be supported stably up to spin speeds of about 3,065 rpm. An upper spin limit of 2779 rpm for this top is observed experimentally and explained as an adiabatic condition. Spin stabilized magnetic levitation is a macroscopic analog of magnetic gradient traps used to confine particles with a quantum magnetic moment.
The Levitron
[1] is a remarkable toy which levitates
in air a 22 gram spinning permanent magnet
in the form of a small handspun top. The top is spun on a lifter plate
on a permanent magnet base and then raised to the levitation height.
The top floats about 3.2 cm above the
base for over 2 minutes until its spin rate declines due to air resistance
to about 1000 rpm. Unlike an earlier magnet
toy which requires a thrust bearing plate to stabilize motion along one
direction[2], the
magnetic top floats freely above the base magnet and is fully trapped in 3
dimensions. (see figure 1) Since Earnshaw's theorem of 1842[3]
rules out stable magnetic levitation for static magnetic dipoles, it was not
obvious to us how the Levitron worked. A simple theory of gyroscopic stability
against flipping proposed by the manufacturer and others[4, 5]
is not sufficient to explain the stability.
Magnetic levitation of spinning permanent magnet tops was discovered by inventor Roy Harrigan who patented it in 1983 [6]. Harrigan persisted in his efforts even after being told by several physicists that permanent magnet levitation was impossible and that he was wasting his time [7]. Besides discovering spin stabilization Harrigan designed a square dish-shaped base that established a suitable magnetic field configuration, made a top with the right rotational inertia, mass, and magnetic moment, found the small capture volume, and invented a means of moving the spinning top to the to the right location. The parameter space for successful levitation is quite small.
Not much happened with the invention until 1993 when Bill Hones of Fascinations learned of Harrigan's patent and saw a working prototype of the levitating top. Hones and Harrigan had a brief collaboration to make and market a levitating top toy but it soon ended [7, 8]. In 1994 Bill Hones and his father applied for a patent on a levitating top that used a square permanent magnet base which was issued in 1995 [4]. The Levitron, made by Fascinations, has a square base magnet with a region of weaker or null magnetization in the center. The Hones' patent states that levitation over a circular base magnet is not possible. We routinely use circular ring magnets which work at least as well as a square base.
Our investigation included measurements of the commercial toy as well as modified experimental versions. We used air jets and then electromagnetic drives to counter the effects of air resistance and to spin the top faster. We also numerically integrated the equations of motion to determine the stability limits and compare to our calculations and experiments. Our most interesting finding is that there is a maximum spin limit beyond which the top is unstable and cannot be confined. Understanding this feature is essential to understanding the actual trapping mechanism.
While writing this paper, we became aware of a paper by Dr. Michael Berry, now published in the Proceedings of the Royal Society of London [9]. He was kind enough to send us a preprint of his paper which we highly recommend. Our conclusions about the trapping mechanism are essentially the same as his. Berry develops the theory of the adiabatic invariant further than we do here. We would also like to thank Dr. Berry for reviewing an earlier draft of this paper and making helpful suggestions.