Stephen A. Lawrence wrote: > My explanation turned out to have a gaping hole in it.
However, I believe I have found some reasoning which implies something like that explanation must be correct, and I've also found a "fix" to it to allow the explanation to go forward, and I think there may even be an experiment which could confirm parts of the idea, at least in general terms. a) COAM -- There is a torque acting on the shaft and the ball bearings. To conserve AM there must be an equal and opposite torque acting on /something else/. What else is there in the system? All I see is the outer bearing race. So, it seems like the mechanism /must/ be an interaction between the outer race and the rest of the system. b) In my previous note (which see for the algebra), I observed that -- The back EMF from the "contact inductance" is - rho*v*I, where "rho" is the inductance per linear unit of the bearing material, "v" is the velocity of a sliding contact (i.e., ball bearing) with respect to the outer race, and "I" is the total current through the race. -- The "forward EMF" from the redistribution of the currents in the outer race as the contacts move around is +rho*v*I, and it is exactly equal and opposite to the "contact inductance"; the two effects consequently *cancel*. c) *BUT* there is another effect: Frequency response! The flip side of hysteresis is slew rate, which is directly related to frequency response. Transformers for 60 Hz current weigh a ton because of their iron cores. Transformers for radio frequencies don't have iron cores. They don't have them, in part, because they don't need them; but in part, they don't have them because iron cores don't work very well at radio frequencies. The hysteresis of the iron means the slew rate of the B field in iron is too low to allow the iron core transformer to respond sensibly to high frequencies. So let's look again at footnotes [1] and [2] in my previous email in this thread, and think about the frequencies involved. In footnote [2], in which the "current redistribution EMF" was found, the current, and hence the B field, is reversing twice as the contact goes around the ring. At 600 RPM, that's a bearing rotation rate, and hence a frequency, of about 10 Hz. No problem there, that's for sure. Now let's look at footnote [1], in which the "sliding contact back EMF" was found. The derivation assumes the current in an infinitesimal piece of the wire switches /instantly/ as the contact slides past, which is, of course, false, unless the wire and contact point both have zero diameter. In the real world everything has thickness, and the current, at these frequencies, is distributed through the depth of the wire. We would expect that the time it takes for the current to change direction as the contact passes by is related to the wire thickness. It might be reasonable to guess that the current actually switches direction during the time it takes the contact to travel a distance equal to the thickness of the wire, at least if the contact area is small enough. With ball bearings, the contact area will indeed be very small. So, if the wire -- or bearing race -- is 2 mm thick, then it might be reasonable to guess that the current switches direction in the time it takes the contact to travel 2 mm. If the bearing is 10 cm in diameter, then this will be about 1/150 the time it takes the contact to go all the way around the ring. That's a reversal; a full cycle would be twice that (two reversals). So, the "frequency" here is about 75 times higher than the bearing rotation rate, or about 750 Hz. It seems like that could be high enough to make the "effective inductance" of the material of the race smaller for the back EMF of a sliding contact than it is for the forward EMF due to the current redistribution as the contact goes around the ring. And, to the extent that that is true, there will be a net forward EMF, which will produce a ring current in the right direction to produce a net B field through the ring which will drive the ball bearings in the direction they're already spinning. Does this really work? I dunno -- one thing that occurs to me is that, if the iron responds "slowly", the back EMF from a sliding contact might have smaller magnitude per unit length but might actually be "smeared" a good distance around the ring, possibly resulting in the total back EMF due to the sliding contact being identical to what I derived below. And in that case this won't do the job. But on the other hand... if it responds slowly, then the fact that the current "behind" each contact is *falling* as time passes will tend to make "slow response" equal to "less response" which, again, might make this effect work after all! And then there's the fact that there are multiple balls in the bearing, and another ball is going to run over the same spot almost immediately, which may cancel the "smearing" effect and so put "slow response" back into the category of "no response". But, well, as I said, on balance ... I dunno. d) Finally, here's an outline of an experiment, which could confirm that the effect is due to an interaction between the outer race and the balls. The key is this: Where's the reverse torque that conserves angular momentum? To see it, pretend we take the wires attached to the outer races, and wrap several turns of them around the outer races. This converts the ball bearing motor to a homopolar motor, and at once begs the question, how is AM balanced? The answer, as with any homopolar motor, is that it's the wires leading to the motor which feel the balancing torque. Typically in a simple homopolar motor it's the wire leading to the center of the disk; in our case, we've basically got two homopolar motors back to back (with two bearings on the same shaft), and it's the wires connected to the outer races which experience the balancing torque. The part of the wire closest to the motor will feel the strongest torque, of course. If the shaft is spinning, then the motor is held in place by the outer races, so the balancing torque is transferred through the support to the Earth. To check to see if the balancing torque is really on the outer races and associated wires, we want to switch to suspending the motor by the /shaft/ and letting the outer races spin. Cut the metal shaft and piece it together with a non-conducting spacer. (Yeah, this is already probably getting awkward, since the rig probably gets hot enough to soften a plastic shaft. Whatever...) Now run some nice heavy wires /straight/ /between/ the outer races of the two bearings, thus connecting them electrically, and also locking them together mechanically. Finally, connect one end of the shaft to the positive terminal, and the other end of the shaft to the negative terminal. The shaft, with its *non-conductive* spacer, now just carries the power to the bearings. The electricity flows along the shaft, out through the first bearing, along the wires between the outer bearing races, in through the second bearing, and along the other part of the shaft to the negative connection. With it all hooked up, give the outer races (and attached wiring) a spin. If the balancing torque is indeed acting on the outer races and/or attached wires, this arrangement of the motor WILL NOT WORK -- it will just spin down as though there was no power attached to it. The reason is that the "power torque" is acting on the balls, in one direction, and the "balancing torque"is acting on the outer races and wires, in the other direction -- but if we're holding the central shaft still, the balls and the outer races must turn in the same direction! To confirm that the bearing "still works" after all the surgery, it would be necessary to cut the wires holding the two outer races together, bridge the nonconducting spacer in the shaft, and wire it up "the old way" and see if it still spins. (And before investing the substantial effort of doing the experiment it would be advisable to triple check my reasoning to see if it really leads to the conclusion I've claimed!!) Unfortunately, this experiment would be a pain in the neck to perform, and if I can judge by the silence, both Horace and Kyle have given up on this particular time sink. If we wait for *me* to do it, we may have a long wait -- though I'm sufficiently fascinated by the question of what makes this go that I may yet take a stab at it.
