“Perpetual motion” is a hypothetical type of motion that continues forever without any external energy input contributing to the system. Students should know that this is generally impossible because of energy losses due to friction or other nonconservative forces, or because some assumption has been made that violates the first or second law of thermodynamics (or both). The first law states that the total energy of an isolated system remains constant over time, and one of the most common formulations of the second law states that the entropy of an isolated system over time cannot decrease. Note that the laws apply to an isolated system (i.e., one that cannot exchange mass or energy with its surroundings), whereas a closed system can still exchange energy even if it cannot exchange mass.1 There have historically been many purported perpetual motion machines.2,3 A close analysis of the machine invariably finds either that there is some assumption that violates the laws of thermodynamics or that there is some input from an external energy source, which means that the system is not isolated.4,5 In this article, we will consider an amusing example of such a machine, in which a ball starts on a platform, falls through a hole, and slides down a track formed from two side-by-side stainless steel rods.6 I explain how the machine can be used in class to illustrate physical principles, outline the possible mechanisms, and end with an explanation of the correct mechanism.

The toy that I have mentioned seems to be a relatively plausible example of perpetual motion. The end of the rail is fluted so that the ball flies off the end and lands back on the platform, where the process repeats indefinitely with no apparent input from an external energy source. Figure 1 shows a view of the device from the side. One can see that the ball slides down a pair of rods that are placed side by side. I initially suggest that students study the apparatus more closely, either via a video such as the one at Ref. 6 or ideally with a physical version of the toy. Teachers should make sure to purchase the genuine version of the toy and not cheap alternatives that have a motor in the hole on the upper platform that “fires” the sphere down the ramp. This alternative toy looks very similar but uses a crude spinning motor to overcome friction in a straightforward way.

Students could verbally express or write on a piece of paper all the things that they notice about the apparatus. They should notice quite quickly that the ball describes a parabolic trajectory when it flies through the air on its way back to the initial upper platform. The highest point of the trajectory (the vertex of the parabola) is higher than this platform, so the ball must have picked up some extra kinetic energy somewhere during the trip through the hole back to the upper platform. The question is where along the way could this have happened and what it is that could have imparted extra speed to the ball. Students should think of how they could potentially figure out where on the trajectory the ball is being sped up. Probably the easiest way to do this is to use video analysis software. A video of a trajectory can be recorded with a mobile phone to be uploaded to video analysis software and analyzed frame by frame. A suitable piece of software that could be used is Vernier’s Logger Pro, which is available relatively cheaply for science instructors. Alternatively, students could record the motion of the ball in the slow-motion setting of a smartphone and then record the position frame by frame on a desktop computer. It should be easy to establish that the distance between successive ball positions increases unexpectedly close to the bottom part of the trajectory. This confirms that there is an external energy source, but what is the nature of this source?

For the purpose of answering this question, I suggest that students should now propose potential mechanisms that explain what is seen and then work through the implications and plausibility of each mechanism. After this, groups of students could argue for or against some of the proposed mechanisms, a technique that reinforces concepts learned earlier in a physics course, but that also requires a deeper understanding of the concepts as opposed to learning by rote or memorization. It would obviously be desirable for students to have access to the toy shown in Ref. 6. The device can be bought online for around a hundred dollars, which seems like a reasonable price for a toy that will be used many times in demonstrations to illustrate physical principles.6 The main benefit of having a version of the device in class is that students can study its motion up close and obtain video images. An exercise that I would suggest is to create a montage of the toy by superimposing between 10 and 20 balls at their locations over the course of one cycle. For example, if T is the time needed for one cycle (around 1 s), then images of the ball could be superimposed at times from T/20 up to T in steps of T/20. This would then allow students to calculate the speed of the ball at different locations along its trajectory and quantitatively confirm the statement above that the ball obtains a burst of speed close to the bottom part of its trajectory.

Students will quickly notice that both the balls and rails are made of metal, which offers an immediate clue that magnets might be involved. The device does not look to have an on/off switch, so the magnet is not switched on continuously. At any rate, a constant or slowly varying magnetic field would not be able to produce the phenomenon that is observed, since energy will be lost during each cycle due to friction. Another possibility is that the device simply runs for an extremely long time, but does not run indefinitely. Obviously, we cannot watch the machine for an indefinite amount of time, but the machine does not seem to be “grinding to a halt” after watching a dozen consecutive cycles. As expected, inspection shows that there is an electromagnet in the base of the device, although instructors may wish not to reveal this to students. Video analysis can confirm that the ball accelerates at the lowest point of the rail, where it is in contact with the top of the base. This lowest point looks to be around two-thirds along the bottom, which matches the location of the magnet in the base of the machine.

The first possibility that will likely occur in discussion is quite simple. Before seeing the electromagnet in the base of the device, students might guess that the magnet is always switched on once the device is started and is not controlled. This possibility can be removed immediately because if it were always switched on continuously, the magnet would take at least as much kinetic energy from the ball as it imparted to the ball. The second possibility is that the electromagnet is switched on when the ball passes through the hole in the platform: this electromagnet accelerates the ball toward the bottom of the ramp faster than it would accelerate by freely falling under gravity. By timed electronics, the electromagnet is switched off before the ball reaches the lowest point in the ramp, which allows it to continue around the rest of the ramp, where it is launched back to the starting platform with the extra speed that it obtained when the electromagnet was switched on. The process is then repeated indefinitely. This explanation is incorrect, however, as the extra kinetic energy imparted in this way would be insufficient to allow the ball to travel all the way back to the starting platform. The presence of a CPU in the base of the device confirms that the actual mechanism is slightly subtler than this and involves sensors.

A related possibility is that the electromagnet is always switched on and pulls the ball down as it is traveling downward, but now the moving ball in the magnetic field creates a change in the current flow that is sensed by a controller in the CPU, which switches the magnet off at the right moment. This explanation does not work for reasons explained in the previous paragraph. Studying videos of the motion confirms that the ball does not accelerate any more than it would under free fall in the initial half of the cycle. This can also be confirmed by dropping several balls through the hole in quick succession such that it would seem to be difficult to correctly time the electronics such that all the balls return to the starting platform.

The correct explanation involves electromagnetic induction. The electromagnet is always switched off and is then switched on momentarily by a sensor when the ball is at the lowest point of the rail. In that case, how is the ball “boosted” by the magnet, since we normally think of magnets as attracting objects toward them, rather than pushing them? The answer to this is Lenz’s law and the presence of eddy currents. When a magnetic field is switched on in the presence of a conductive object, an eddy current (or Foucault current) is induced in the object. An eddy current is a loop of electrical current induced in a conductor by a time-varying magnetic field in the conductor. This process is governed by Faraday’s law of induction, also known as the Maxwell–Faraday equation or Maxwell’s third equation.7 

Electromagnetic induction is the effect whereby a changing magnetic field induces a current. A well-known example is a bar magnet moved through a stationary wire loop to induce a current around it. The “change” can be relative, so it could be the magnet moving through the loop or the loop moving back and forth along a magnet that is at rest. More specifically, time-dependent magnetic fields are coupled with nonconservative electric fields. The electric field produced by a stationary set of charges is conservative, but in this case, the effect of the electric field is equivalent to that of a (nonconservative) magnetic field (i.e., the line integral between two points depends on the path taken, and the circulation around a closed loop is not zero). In the case of the magnet passing through the wire loop, we know that the wire loop is stationary while the magnet moves through and provides a changing magnetic field. A current is produced, so an EMF (electromotive force) must be driving it. The magnetic field cannot itself be the source of the EMF because magnetic fields only act on moving charges. Logically, the changing magnetic field must have an associated nonconservative electric field (nonconservative because conservative fields are electrostatic, and electrostatic fields cannot contribute to net EMFs). The induced EMF around a closed curve is the same as the voltage drop (i.e., the same as the line integral of the electric field around that curve). Mathematically, this is encoded by Maxwell’s third equation in integral form:
CEdl=ddtSBdS,
where E and B denote the electric and magnetic fields, respectively. In other words, the induced EMF around a closed curve is equal to the rate of decrease of magnetic flux over an open surface that has that same curve as its boundary. If one uses the curl theorem, the same equation can be written in differential form as
×E=Bt.

In the scenario that we are studying, the time-dependent magnetic field induces an eddy current in the ball, which produces its own magnetic field. Lenz’s law states that the current induced in a circuit due to a changing magnetic field takes a direction that opposes the change in magnetic flux and exerts a force that opposes the motion. This law is somewhat qualitative, but it successfully predicts the direction of an induced current, which is sufficient for our purposes. Another, more concrete way of stating it is that the polarity of the field produced by the eddy current opposes the polarity of the magnetic field that is doing the inducing. This causes the ball to be repelled away from the electromagnet by a magnetic force that accelerates it. The end of the rail is then cleverly designed so that the ball flies off the end and lands back on the starting platform, where the process can repeat itself again indefinitely. The description that I have proposed is confirmed by online descriptions of the product, which describe the toy as being a “magnetic induction” perpetual motion simulator.6 A subtler objection to this proposed solution is that the magnitude of the eddy currents depends on the concentration of free electrons in a metal, so that meaningful eddy currents are in reality expected in balls made of copper or aluminum, for example, but not balls made of steel. However, the material used for the balls is described in the online product description only as “metal” rather than stainless steel, whereas the rails are explicitly described as being made of “stainless steel.”6 

Finally, I observe that this investigation has been mainly theoretical and that there are a few remaining questions that could be resolved empirically by students with access to the device. The motion of the ball and the way it “jumps” due to Lenz’s law is quite similar to a classic demonstration known as Thomson’s jumping ring, where an aluminum or copper ring is placed on the end of an electromagnet. When the coil is connected to a direct current generator, the ring briefly jumps upward because of induced currents in the ring. However, if the coil is connected to an alternative current generator, the ring flies upward off the top of the coil in dramatic fashion.8,9 Explaining why an AC current has this effect is not an easy exercise.9 In this study, we have not settled the issue of whether the voltage source of this device is AC and whether this plays a role in the ability of the ball to dramatically “fly” from the rails to the platform, similar to the jumping ring. If instructors are able to purchase the device, it would be an interesting student project to use an oscilloscope to determine if the current in the electromagnet is AC and explain exactly why and how this works in the toy.

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Hollis Williams is a postdoctoral researcher at King Abdullah University of Science and Technology. He is interested in various aspects of physics education and theoretical physics and has published articles on fluid dynamics, quantum mechanics, and particle physics.