Max

Good example, but I'm sure Tony doesn't understand what's happening here.

Exactly, you got it.

You may have mentioned it, but your argument was completely wrong. Running boys in a circle has nothing to do with rolling circles. As I've shown, it is the combination of translation speed and tangential speed that explains that the smaller wheel is slipping.

Yes, there is slippage, that is what Aristoteles was missing. Ah, rewriting history... You don't understand what Aristotle wrote, and therefore "some other person" (not too bright presumably) must have added the second track.

Impossible, as I've proved many times. Replace your wheels by gearwheels (also forming one solid wheel) and the surfaces by corresponding racks, then you'll have perfect grip between gearwheel and rack (ensure that the gears cannot leave the racks). Now you'll observe that these wheels cannot roll at all. That is reality! Just try it if you don't believe. Explain why it is impossible for the gearwheels to roll. Hint: it has something to do with the fact that the small wheel is unable to do something, thanks to the perfect grip of the gear system.

False, the crux of the paradox is that both wheels cannot do a "true roll" without slippage. Reality is that the smaller wheel is slipping, reality is not what you're imagining. Aristotle brought that second track in, your suggestion that that is some newfangled invention of ours is disingenuous, we just keep to the original formulation! Further, nobody claims that that wheel behaves differently when this track is "brought in", it only is a reference that makes clear that the smaller wheel is not rolling out its circumference, but makes another movement that we call slipping.

I can't remember, but you stated your position again so explicitly, that I was wondering why it should be so important. As you'll have seen, there are many different definitions of a paradox, and also many different kinds of paradoxes. No problem for me, I don't believe so much in the "one and only" correct definition à la Rand (e.g. her definition of altruism). My viewpoint is, that so many of those classic "paradoxes" are known as "paradoxes", that I see no reason not to use that term for that kind of "paradoxes", genuine or not. For myself I use the definition: an apparent contradiction in an argument caused by a more or less hidden error in the argument or in the premises. In general it isn't difficult to move the error from a false premise to an error in the argument, and an error in the argument can always be thought of as the result of an implicit false premise, there is no sharp distinction between the two options. Changing the formulation a bit can change the formal expression of a paradox, without really changing its essence. Therefore I think my definition isn't that much different from yours, only less restricting, while I also admit false premises. But as I said, I find definitions not that important (the only correct one!) as long as you state them clearly.

But that doesn't solve Aristotle's paradox, it's avoiding it by changing the conditions. That there is no paradox when two concentric wheels can rotate independently from each other, Aristotle no doubt could have also figured out, but that wouldn't have helped him solving his paradox. The two wheels forming one rigid body can easily be realized in a physical system, no contradiction there. The contradiction emerges when you suppose that both wheels can roll without slipping/both circles can trace out their circumference at the same time. That is the essence of the paradox.

Why should I? Did I deny anywhere that the internal wheel traverses a distance greater than its own circumference? That has always been part of my argument. It can only traverse the distance of the outer tyre's circumference if it slips. Otherwise it is impossible. No vague suggestions, show the calculations, just as I've done. I've also shown that it is just the differing tangential speeds that explain the slipping. Reality is not what you think it is, but what it is.

Wrong, it is rolling and slipping. And that is the definition of slipping. If it wouldn't slip, it could not traverse a distance greater than its own circumference. Show me how the wheel could traverse a distance greater than its own circumference without slipping, and without resorting to pure magic or meaningless objectivist buzzwords.

Really? Rolling one revolution, far past its circumference: that is by definition slipping, it is definitely not rolling without slipping, because then it couldn't get past its own circumference.

Peikoff parrot. He also started to talk about "floating abstractions" when things went way over his head.

I think you're right. We have now posted so many diagrams, animations, videos and mathematical derivations, that anyone who is seriously committed to finding the solution to this puzzle has material enough for studying the problem, to either accept the slippage solution, or to come up with a valid counterargument. showing what would be wrong in our examples and derivations and to give an alternative solution, and not a "solution" that isn't. The people who don't get the original problem, won't get this version either. Bob had directly the solution of the original version. I think he's still more or less thinking of that one, not yet seeing that the new puzzle is different (though a variation on the old theme). So I wouldn't bother too much about this version, he'll get it sooner or later. Indeed, this is where the men are separated from the boys.

What highly abstract reasoning? It's all rather simple and the math is also quite elementary. I think that a big problem for those old guys was that they didn't know calculus, they didn't for example have a notion of the concept "instantaneous speed" (Zeno's arrow problem!), while that is now a piece of cake for us. So they were puzzling about wheels jumping over gaps in their supports, trying to make sense of it all.

The point is that the inner wheel cannot be dragged, as it is held back by its own cable (that is fixed to the support). It cannot roll further than r*theta, its cable is taut. From its attachment point on surface 1 then a piece of string with length r*theta lies stretched on surface 1, the rest ist still wound around the wheel. Perhaps it helps to look at the figure: Therefore it is now the small wheel that determines the movement, the large wheel must "follow", that is, it is held back, slipping, while it rotates together with the small wheel. The large wheel has unrolled R*theta of its own cable, while it has only traveled over a distance of r*theta, there is "too much" unrolled cable from wheel 2, therefore it is slack, and lies there like a dead snake when you roll far enough. I suppose Jonathan is now making a new picture or animation to make it even clearer...

So according to you, Zeno's paradoxes, the twin-paradox, the barn-pole paradox, the bug-rivet paradox, the Gibbs paradox, Olbers' paradox are not genuine paradoxes, although these are well-known as such?

...and explain it away, telling us that they've "solved" the paradox.

It is telling that the people who don't accept the slippage explanation of the paradox apparently feel compelled to remove essential elements from its original formulation. Why would that be so?

That is half the story. The other half is the fact that in this case the large wheel cannot "roll without slipping". That would namely imply that the smaller wheel would be dragged along, slipping to keep up with the large wheel (as we've already shown in about 10000 posts). But the small wheel is held back by its shorter cable, so that's the only wheel that can roll without slipping. That again causes the large wheel to slip: it is held back, rotates more than its "rolling distance", causing its own cable to become slack.

It's the other way around. The shortest cable determines the movement, as it can't be lengthened, but the large cable can be loosened. The small wheel rolls without slipping, generating a proper cycloid. The large wheel is slipping backwards, loosening its cable, generating part of a prolate cycloid.

We don't know for sure that Aristotle wrote that text at all, perhaps it was Archytas of Tarentum, as has been suggested. But whoever it was, we can't know whether he was looking for solutions or not. I think he was (it would be rather unnatural for such a person not to try to solve the puzzle), but that he couldn't find the solution. After all, after him people like Galileo, Mersenne, Fermat and Boyle also tried to solve the puzzle.

I've explained that in detail in this post (click on the arrow): Didn't you read that? It answers all your questions. And you'll also see why these are silly questions.

Observe that it is just the fact that the tangential speed of the large circle is greater than that of the small circle, is the cause that the lower part of the small circle moves faster to the right than the lower part of the large circle, as the horizontal component of the tangential vector has to be subtracted there from the translation speed, and subtracting a smaller value results in a larger speed than subtracting a greater value!

That is true in the rest frame of the circle, the tangential speed of the outer circle is greater than the tangential speed of the smaller circle. But wait! We are considering the system in the rest frame of the track, where we see the wheel rolling to the right. In that frame you have to add the translation speed to the speed of the points on the circles. Due to the rotation, a point on the large circle continuously changes direction. In the lower half of the figure the horizontal component of the velocity vector of that point is directed to the left. So we have to subtract that horizontal component from the speed due to the translation to the right. In our rest frame, the point is moving slower than the center. In the 6 o'clock position the tangential velocity vector is exactly directed to the left. The speed in the rest frame (subtracting now the tangential speed from the translation speed) zero. At that one moment the point stands still. That is equivalent with the condition "rolling without slipping". *)Further rolling of the circle decreases the horizontal component of the velocity vector, so the speed in the rest frame increases again. In the upper half of the figure the opposite happens. After passing the 9 o'clock position the speed becomes greater than the translation speed of the center.At the 12 o'clock position the velocity vector points to the right and now the tangential speed is added to the translation speed, the point has now a speed twice that of the center. Logical, because after one revolution every point on the circles must have traveled the same distance to the right, so what they lose in the lower half, they must make up for in the upper half and vice versa. Now look at the small circle. When the segment AB of the large circle lines up with CD of line 1, around the point of zero speed, you see that the corresponding segment EF of the small circle is swept to the right along a much larger segment GH of line 2. If the small circle would roll without slipping, like the large circle, it would in the same way line up with an segment GH that is just as small as EF. But as the tangential speed of the smaller circle is smaller than that of the large circle, the amount that is subtracted from the translation speed is smaller, and therefore it doesn't cancel the translation speed at that point (as in the case of the large circle), therefore instead of zero speed, there is a net translation to the right. That net translation we call slipping, and it is very well visible in this animation. *) For cycloid lovers: this is the point where the cusp of the cycloid touches the line.

But I showed you that the concept is present: Unrolling the large circle to the line ZI means that ZI is the track over which the large circle rolls, and unrolling the smaller circle to the line HK means that HK is the track over which the small circle rolls As if that matters. Ever heard of technology? Of people who make machines, cars, airplanes, bridges, cranes, etc. etc.? Where everything depends on the fact that we can use mathematics to reliably calculate forces, distances, stresses, angles, speeds, torques etc.? Do you think they're worrying about the question whether a line on a technical drawing could be the representation of a track, or that it is more like an imaginary path? Miraculously, airplanes built based such drawing boards with imaginary paths can fly! That is reality! Nonsense. I've shown that a completely mathematical treatment also gives the slippage solution. As you'd expect from a correct solution, a physical realization confirms the fact that slippage occurs. Theory and practice are in agreement. You are the one who is destroying reality, by refusing to see what nearly everybody can see, and refusing to consider the mathematical treatment. Two tracks it is. Why do you think Aristotle (or whoever that old Greek was) wrote about two lines, along which the circles unroll. Just to make a pretty picture? No, that was because those two lines are essential to the formulation of the paradox. If you don't understand that, you don't understand the paradox. And that what happens is called slipping. When you realize that, all strangeness disappears and the paradox is solved, as it was generated by the assumption that the smaller wheel could also roll without slipping - mathematically: the false assumption that the smaller circle could also trace out its circumference by rolling one revolution.