Max

The problem is then stated: ----- J You beat me to it... That the word "track" isn't mentioned there, does of course not mean that the concept "track" is absent. 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. So those two tracks are an essential part of the original paradox. Taking those away is destroying the paradox, not solving it. Child and bathwater.

But the tangential speed for different points on a circle is the same. It is the speed in the rest frame of the circle, no translation. Of course it is different for points on circles with different radius, perhaps that's what you mean. The original paradox was stated in terms of rollig circles. Any problem with circles is a geometrical/mathematical problem, so I don't see why a mathematical treatment of the paradox would not be the ideal method to solve it. Those circles are tracing out their circumference, a corresponing physical object would be a wheel, rolling without slipping it is the equivalent of a circle tracing out its circumference. Any objections so far? Now a wheel is a very good object in this case, as wheels are meant to roll without slipping, and a wheel concentric in a wheel (just as a circle within a circle in the original description) is easily realised (flange, hub), so a practical test of Aristotle's paradox is fairly easy to realize. Now about your archery target: perhaps it isn't difficult to rotate it, but that isn't yet rolling. For that you'd have to accurately cut out the target at the outer circle and roll it over the ground or some other support. But then you still haven't one of those smaller circles rolling. Rotating, yes. But rolling needs a support and that circle has to be raised from the rest of the target to allow contact with that support. Now I seriously doubt that you've done that. Probably you just imagined that doing, but that is not good, reality-based evidence! Especially as you apparently already have great difficulty in observing the slipping in the animations and videos that we've seen here, and where many people clearly see the slippage. Some objects are just much better to visualize some effect than other object. The iris and pupil of the eye for also two concentric circles, but they are not well suited for a demonstration of Aristotles paradox. How would you roll an iris and a pupil? Yes in you imagination, but then you'd better concentrate on the mathematical solution. Experiment and mathematical analysis show definitively that slippage occurs on the smaller wheel, if the large wheel rolls without slipping. It that is not basing it on observing reality... But you do need them with circles, those are the equivalents of the tangents that form an essential part of the original problem. You may take them away, but then you take the problem also away. Child and bathwater!

First, you should define your terms accurately. Velocity is a vector, speed a scalar. Rotational speed ω for a point on a circle with radius r is the number of revolutions/time unit. Linear speed is the distance traveled/time unit. For a circular motion, linear speed = tangential speed = rω, proportional to r. This is trivial. The rotational speed is for both boys the same, but boy A has a larger tangential speed. But this is not relevant to the problem of Aristotle's rolling wheels. In this case the rotational speed is for both wheels the same, the tangential speed is for the small circle r/R smaller than for the large circle. But the translational speed, defined as the distance traveled by the center of the wheels after one revolution, is the same for both wheels, and equals 2*pi*R if the large wheel rolls without slipping (essential condition). The smaller wheel would travel 2*pi*r if it also rolled without slipping, but that is in contradiction to the fact that it travels a distance of 2*pi*R. The difference must be made up by the small wheel, and that implies that it must be slipping against its tangent. It cannot roll without slipping, like the large wheel. Clear? Silly question. Dit they roll?

Those transition-cycloids may be interesting in themselves, but they don't contribute anything to solving the paradox. Perhaps you're inspired by the cycloids in Mr. Drabkins's book, but these don't give a solution either. Sidestepping my point. You were berating me for not making a distinction between slipping and skidding. I pointed out that the authors of the link you recommended don't distinguish between slipping and skidding either, that they used slipping, positive or negative, for both cases. And the point about using scare quotes: the small wheel/circle is also slipping against its support/tangent in the original article, so there is no reason to use scare quotes. That you insist on removing that tangent from your "solution" doesn't make it disappear.

Sez who? Aristotle himself? And who wrote that section? But talking about Aristotle's wheel paradox is no problem? It is allowed translate one circle as a wheel, but not the second circle? Sez who? That is what happens with a real wheel. In the idealized continuum case we should take the limit → 0 of a segment of the circle touching a larger segment of the tangent. But why should we accept the authority of Drabkin? I think you should add this link to the Wikipedia page: https://www.humanities.mcmaster.ca/~rarthur/articles/aristotles-wheelfinal.pdf for a good discussion of the paradox and Drabkin's ideas.

Ever heard of the concept "irony"? Didn't you realize that I was just paraphrasing a post of your comrade Anthony? Tsk tsk!

This is your original post I reacted to: Observe the strange logic: 1. Since the center's path is 2*pi*R, every cycloid of the smaller circle is greater than 2*pi*r, i.e. its circumference. Every cycloid of a circle with radius r is greater than 2*pi*r (unless in the degenerate case when you force the circle to remain in the same place, then the cycloid is identical with the circle itself), this has nothing to do with the fact that the center's path is 2*pi*R. 2. Therefore, the smaller circle rolls farther than it would by pure rolling. So the fact that every cycloid of the smaller circle is greater than 2*pi*r implies that it rolls farther than it would by pure rolling? That is what you write. Now you claim that you meant that the fact that the circumference of the small circle is smaller than that of the large circle implies that the small circle rolls farther than it would by pure rolling. But that is not what you wrote! Ironically you write now that it is "not about a cycloid", but you are yourself continuously talking about cycloids in your "argument". So you admit now that these cycloids have nothing to do with solving Aristotle's paradox!

Dunning-Kruger

Special pleading? Heh, I'm just discussing the original version of the paradox, with two tracks and two wheels that roll without slipping, not some fantasy of yours.

That guy was that idiot Aristotle, who even didn't know the identity of the wheel! He was of course so stupid that he overlooked the internal logic of a wheel. It's clear that he must have never seen a wheel in his life.

Of course you should explain mechanically or by math what is a mathematical/mechanical puzzle and nothing more. That "perceptual-conceptual exercise" exists only in your imagination. No, I have the paradox premise exactly right. The premise is stated in the original version of Wikipedia: "The wheels roll without slipping for a full revolution". I have proved that this is false and that the fact that it is false, is the origin of the paradox. That we all use 2 tracks is because these are necessary, as the statement "wheels roll without slipping" would be meaningless otherwise. Those 2 tracks are not some invention by us, they are given in the original description!! And how do you think that it is possible that the small wheel travels a larger distance than its circumference? Drum roll..... Right! By not only rotating, but also slipping!

The joke is of course that the fact that this premise is false (both wheels cannot roll without slipping) is the cause of a paradox emerging. You can solve the paradox by showing that it is impossible that both wheels roll without slipping and why that is so. It is like a magic trick: the magician tells you something or suggests you something by his actions that is not true, but that the naive onlooker accepts for true, which makes the next actions of the magician seem to be impossible. A general recipe for creating a paradox is to tell a plausible story, with a plausible argument, but with a hidden error in the premises or in the argument, leading to an apparent contradiction. So the fact that the premise of two wheels rolling without slipping is false is essential for the existence of the paradox, otherwise there wouldn't be a paradox at all!. Further, if you can't be convinced by the (excellent) videos and animations in this thread, you can always check the mathematical derivation I gave, it is quite simple. Avoiding it is avoiding reality.

A good example of the nonsense you get with all that talk about "identity". You apparently think that painted archery rings are essentially the same as car wheels, just while both can approximately described as concentric circles, and that the fact that archery rings usually don't roll or slip "proves" that car wheels can't roll or slip (let's hope you don't drive a car). If you want to use the abstract property that archery rings and car wheels share, then you should concentrate on circles. Circles can roll and slip, as I've several times demonstrated, even if painted archery rings can't. Of course Aristotle chose a mechanical implementation with wheels, as these are a natural implementation of rolling circles, that are at the core of his paradox. It is a mechanical/mathematical puzzle, that of course can and should be solved by mechanical/mathematical reasoning. And the mathematical solution isn't complex at all, it is in fact very simple. Well, except for people who insist on "identification". Where is that "identification" now? Does it come soon?

From the original version of the Wikipedia article: "The wheels roll without slipping for a full revolution." Rolling wheels, you know. No doubt chosen by Aristotle while it is rather natural for wheels to roll and to slip or not to slip, in contrast to rings of an archery target. Does the fact that rolling and slipping of archery target rings is a rather silly notion imply that rolling and slipping of car wheels or train wheels is ridiculous?

I don't understand that. I see no difference between "the second wheel is along for the ride" and "two wheels stuck together". In both representations there is only one rigid object, that consists of two wheels with a common center. The only difference in interpretation would be either to treat those wheels as a mechanical system or to treat the paradox as a mathematical problem, but as I've shown in earlier posts, the two descriptions are equivalent.

I don’t think he meant that, after all he also writes “...greater than 2*pi*r… with small r, so he’s referring to the small wheel. But even if he meant 2*pi*R, his conclusion doesn’t follow: the cycloid of the large wheel is also greater than its circumference, but the large wheel is rolling without slipping (tracing out its circumference), so this condition is no guarantee for rolling farther than its circumference. But he doesn’t find that, as his cycloid argument is fallacious (see above). It is really worse than you think...

In my post of Januari 26 I said that I don‘t make a distinction between slipping and skidding. And I‘m not the only one. According to your view, everyone who writes “rolling without slipping” is wrong, as it then should be “rolling without slipping or skidding”. Better even, look at the previous link you gave yourself: click on “summary” and scroll down to “rolling and slipping”: those guys don’t make a distinction between slipping and skidding either, they call it all “slipping” positive or negative. And they don't put "slipping" between scare quotes! In a link that you recommend! Ever seen a train wheel? Anyway the point is moot, as Aristotle’s paradox is not about the practicability of wheel designs, old Greek or modern, it is a thought experiment. The only important point is that the system can in principle be built in reality (which will show that the condition that both wheels can roll without slipping cannot be met). In the original Wikipedia article there was also a line/surface drawn under the small wheel. It is not a crutch, it is the crux of the paradox, as the paradox description states that both wheels roll without slipping. Whether such a surface exists in reality (it does for train wheels) is irrelevant. If you like you can avoid Aristotle's mechanical language and translate into mathematical terms. Then you can substitute “traces out the circumference of the circle” for “rolls without slipping”. You wrote: Summarizing, the smaller circle moves horizontally 2πR because any point on the smaller circle travels a shorter, more direct path than any point on the larger circle. That “because” here is nonsense. The small circle moves horizontally 2πR because the large circle moves 2πR, and those circles have a common center, which implies that the small circle must slip, oh sorry, I mean “does not trace out its circumference”, as was implied in the description of the paradox. Further, without the latter point, you have not solved the paradox, you've only told us that both circles moved the same distance, what we knew all along, as part of the paradox description. That those cycloids are completely superfluous, you show yourself in your second “solution”, in which no cycloid at all is mentioned. No, we are the four horsemen of the apocalypse. Well, three horsemen and one horselady.

Applying the same argument to the large circle: every cycloid of the large circle (cycloid length = 8*R) is greater than 2*pi*R . Therefore, the large circle rolls farther than it would by pure rolling? You've created a new paradox! At least I'm not the one who is drowning.

When I saw your previous video, I was sure that this would be the next one, all the attributes were already there...

The diagram is in so far incorrect, that it doesn't represent a wheel that is rolling without slipping (which was the supposition in the description of the paradox): the distance traveled after one revolution is smaller than the circumference of the large circle. But apart from that, it is a completely valid diagram, it's perfectly possible that both wheels are slipping. No, that line doesn't represent the movement around the circumference of just one of the circles, it just marks two points on those circumferences, thereby forming a mark for the amount of rotation. You could very well paint such a line on a real wheel, and it would rotate exactly the same way. Those two intersection points rotate completely synchronously. However, a different thing is that at least one of those points is also slipping along its tangent line. You can see that also in slow motion.

In the 17th century he would have told Newton that Math formulations (of planetary orbits around the Sun) were *derived from* the identity and causal actions of sun-cum-planets, they can hardly be used *to prove* any controversial aspects - assuming one thinks circular reasoning is invalid, Newton's calculations like the derivation of Keplers laws was therefore invalid. Circular reasoning! Or perhaps elliptical reasoning? How does he think that the "identity" of something is determined? By divine relevation? Or by Peikoff speaking ex cathedra? It's totally hopeless, a rational argument with him is impossible.

Yes, it will slip, as I've proved mathematically (see my post of February 4) and as Jon and Jonathan have very clearly visualized in their videos and animations (do you insinuate that these are optical illusions?). Show me were I made an error in my proof, if you can, and with real hard arguments, not with some confused metaphysical nonsense like "self-contradiction to the identity of the wheel".

Indeed. And it's a pity that he's messed up this Wikipedia page with those useless cycloids, giving "solutions" that aren't.

>> = me > = merjet black = me > >So you admit that it is a solution." > Wrong. I said you believe it, not I believe it. No, that is not what you said. You said: > I get it. You believe there is only one correct solution -- merely because you like it. There are many proofs of the Pythagorean Theorem. Is only one of them correct merely because you like it best? That implies that you believe that there is more than one correct solution, and that implies again that you admit that the mentioned solution is correct, otherwise you’d said “you believe the wrong solution”. >> Therefore the notion of slippage is essential for understanding and solving this paradox." > Wrong. I gave two solutions, neither of which invoke slippage. Also, the Wikipedia article says the larger circle rolls without slipping. Get it? No slippage! The fact that the article mentions that the larger circle rolls without slipping is already an indication that the notion of slippage is important (why mention it otherwise?). Moreover, the original Wikipedia article you linked to in your first post stated: The wheels roll without slipping for a full revolution. That was much better than your new version, as it showed immediately the crux of the paradox, namely the supposition that both wheels at the same time can roll without slipping, which is impossible. >> "Wrong. As I’ve shown above, the “translation solution” isn’t a solution, it’s just stating one half of the paradox problem." > Wrong. The problem as stated says nothing whatever about the necessary properties of translation nor even mentions translation. As if the word “translation” would be essential! The article mentions “The distances moved”, which of course implies translation. The paradox is that those distances seem to be different while they must of course be equal. That story of the cycloids is completely unnecessary, as the only thing you conclude from that story is that the center(s) of both wheels travel over the same distance. You don’t need any cycloid to “prove” such a trivial thing. In your second “solution” you just state the same, without the whole cycloid circus. Further, you still haven’t really solved the paradox, you’ve only “proved” that those two distances must be equal. Well, we knew that all along, as that is part of the paradox statement. What still has to be shown is, why the supposition that both wheels roll without slipping (explicit in the original version, implicit in the new version) is wrong. The answer is of course that when one wheel (the large one or the small one) rolls without slipping, the other wheel automatically must be slipping to travel the same distance, to make up for the difference in traced circumferences. >> "You can’t expect me to “answer”, as you didn’t ask me anything in that post." >Wrong. I quoted your using "support" and I asked "Its support??" Then I repeated a formula from August 6 challenging anybody who read it to quantify the three terms on the right side of an equation, shown again below. I'll even pre-fill the 3rd term for you with what you have asserted. That is disingenuous, “its support?” is not a serious question. You meant apparently those formulae, but there you didn’t ask anything. Sorry, but I can’t know what you’re thinking if you don’t express yourself clearly. > 2*pi*R = Rotation + Translation + Slippage 2*pi*R = _____ + _____ + 2*pi*(R – r) Fill in the blanks. Is that so difficult? Not at all. First, those variables are not independent, we're talking about the (center of the) circles, not about a point on the circles. For the large wheel (radius R) that rolls (one revolution) without slipping: Translation = Rotation = 2πR For the small wheel (radius r) that rolls and slips, while the large wheel rolls without slipping: Translation = rotation + slippage = 2πr + 2π(R-r) = 2πR, as you’d expect. Simple comme bonjour!

Nice example of an instructive illustration with very simple means. It's sooo obvious!