PenduSMOT
Introduction
Recently, there has been some speculation involving a pendulum with interacting permanent magnets (known as the PenduSMOT). The configuration is shown below in Figure 1. A small bar magnet is mounted on the periphery of a rotating disk. Below the rotating disk, a large, axially magnetized permanent magnet is placed in a fixed position. The rotor orientation shown in Figure 1 is assumed to be the nominal position. Behaviour that some suppose to be anomalous occurs when the disk is rotated 900 from the nominal condition and released. Depending on the direction of the initial 900 displacement, the rotor swings to a position on the other side that is consistently either higher or lower than the initial magnet position. Some believe that when the magnet achieves a higher position than when it was released, it represents an anomalous "energy gain." This note will calculate the magnetic and gravitational energy for a particular configuration in an effort to shed light on this behavior.
Figure 1: PenduSMOT Configuration.
Configuration
The configuration under consideration has a 1/4" diameter, 1" long magnet mounted to the spinning rotor. The center of mass of the bar magnet is located at a distance of 4" from the center of the axle. The stator magnet is 2" in diameter and 1/2" thick. The center of mass of the stator magnet is fixed 7" below the center of the axle. Both magnets are N40 material (NdFeB magnets with an energy product of 40 MGOe). For the purposes of analysis, it is assumed that these magnets have a coercivity (Hc) of 1 MA/m and a remanence (Br) of 1.25664 T. The magnets are assumed to have a density of 7.4 gm/cm3. For the purposes of this analysis all dissipation (such as friction in the bearings of the axle, air drag, and eddy currents induced by the motion of the rotor-mounted magnet) is assumed to be negligible.
Analysis Tools
To analyze the magnetic field energy, the free 3D magnetostatic field solver Radia was used. Radia uses Mathematica as a user interface; a Mathematica notebook that analyzes the PenduSMOT is available as PenduSMOT.nb. Radia computes magnetic field energy by essentially repeatedly applying the expression for energy of a magnetic dipole (see http://en.wikipedia.org/wiki/Dipole), adding up the the interactions of all the parts of the stator magnet with all of the parts of the rotor magnet. The configuration is simple enough, however, that a nearly identical result can been obtained by idealizing both magnets as point dipoles, a method which yields an closed-form expression for magnetic potential versus rotor position. (A version of the notebook idealizing the magnets as point dipoles is available as PenduSMOT_Dipole.nb, with a print-out of the notebook available as PenduSMOT_Dipole.pdf.) The magnetic potential energy obtained by Radia, along with the gravitational potential energy of the disk, and the sum of the two energies, is shown below in Figure 2.
Figure 2: Potential energy versus angular position.
Interpretation of Results
It is interesting to note that magnetic energy has odd symmetry and gravitational energy has even symmetry. The sum of the two potentials, however, is asymmetric. It is the asymmetry of the combined potential that leads to the "interesting" behavior.
Stable Rotor Positions
The local minima in the total potential represents stable spots at which the rotor could rest. Since there are two local minima, the rotor could be placed stably at rest at either of these spots (at -16.90 and +44.50).
Angular Extremas
Since there are no sources or sinks of energy, the sum of magnetic potential energy, gravitational potential energy, and kinetic energy must add up to a constant value. That constant value is the sum of the magnetic and gravitational potential energy energy at the instant which the penduSMOT was released (assuming that no kinetic energy is imparted to the disk during the release).
If the objective is to find the starting and ending points of a swing, it can be noted that the starting and ending points coincide with the points where kinetic energy equals zero. The height reached on the "other side" of the swing can be calculated by searching for that angle on the "other side" of the pendulum that has the same potential energy as the release point. It is possible to compute the ending position of the swing graphically drawing a line of constant energy from the energy at the starting point and finding the intersection with the "other side" of the curve, as shown in Figure 3.
For this particular geometry, if the rotor magnet is dropped from an initial position of +90 degrees, the analogous "other side" position with the same total potential energy is -95.1392. The system apparently "gains" gravitational energy, but it does this because the magnetic potential energy is less--potential energy is strictly conserved. At points between the two extremes, where potential energy (shown in Figure 3) is less than at the extremes of motion, energy is stored in the kinetic energy of the disks's rotation. If it is assumed that the rotor disk is 9" in diameter, 1/4" thick, and composed of G10 with a density of 1800 kg/m3, the rotor angular velocity required to conserve the system's initial energy (i.e. sum of gravitational potential energy, magnetic potential energy, and kinetic energy = a constant) is shown below in Figure 2.
Going the other way, if the rotor magnet is dropped from -90 degrees, the analogous location with the same total potential energy is +83.8252. Considering just the gravitiational stored energy, this result would appear to be a loss of energy, but again, the total potential energy (the combination of magnetic and gravitiational potential) is strictly conserved.
Figure 3: Constant energy lines for releases at +90 and -90 degrees.
Figure 4: Angular velocity versus position for pendulum swinging between +900 and -95.10.
It is also interesting to note which side is predicted to swing "higher." Objects tend to move from regions of high potential energy to regions of low potential energy. Looking at the magnetic potential energy, it is straightforward to see that the magnetic potential is repulsive (tending to push away from the stator magnet) at +900 and attractive (tending to pull towards the stator magnet) at rotor positions near -900. The mathematics predicts that the magnet will swing from a position on the "repulsive" side to a higher position on the "attractive" side.
Conclusions
All of the behaviors predicted by applying conservation of energy have been observed by amateur experimenters in practice:
- Two stable rest positions of the rotor
- Asymmetric swing height
- Achieved swing height higher when released from the "repulsive" side of the magnet and swinging into the "attractive" side fo the magnet.
Rather than representing a violation of convervation of energy, PenduSMOT serves as an eloquent demonstration of the conservation of energy in practice. A number of behaviors which, at first, may be seemingly odd, are "natural" results that can be viewed as direct consequences of conservation of the total energy in the system (magnetic+gravitational+kinetic).
Appendix: Magnet Flipping
It has also been suggested that there could be a some sort of anomalous behavior if the gravitational potential energy "gain" on one side of the swing is used to flip the rotor magnet at the end of its swing. The same analysis methods used in this note predict that there is no possible gain to be had by passively flipping the magnets.
The analysis tools above assume a model where the magnet can neither source nor sink energy, and in which the total energy is always constant (dissipation mechanisms like friction, air drag, eddy currents, hysteresis, and so on have been neglected to make the analysis simpler). One can therefore summarily conclude that, with the same magnet model and physics, using the "increased" GPE as a "source" of energy to flip the magnets won't increase the total energy of the system (because that behavior was assumed at the outset).
Specifically, consider the swing from +90 degrees to –95 that's discussed above. If there were no dissipation and no "flips", the rotor's predicted trajectory is:
Swing: +90 to –95
Swing: –95 to +90
Swing: +90 to –95
Swing: –95 to +90
and so on…
Some passive device ("passive" in the sense of neither adding nor subtracting energy from the system) could be made that used the "extra" height to flip the magnet. However, the total energy would still be conserved at all times (because there's no mechanism in the system to source or sink energy). It would "take" some height to do the flip—the GPE would be exchanged for MPE. As shown in the Wikipedia dipole article, flipping the magnet's direction by 180 degrees flips the sign on the energy. Flipping the magnete flips the curve in Figure 2 (trades + angles for – angles). The predicted behavior would then be:
Swing: +90 to –95
Passively flip magnet: –95 to –90
Swing: –90 to +95
Passively flip magnet: +95 to +90
Swing: +90 to –95
and so on…
The swing amplitude of the rotor could be made to increase by actively flipping the magnet, i.e. if the device doing the flipping were supplying the power to perform the flip, without letting the rotor magnet drop during the flip. However, this is still no violation of CoE, because the additional energy is provided to the system via the actuator that is performing the magnet flipping.
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