Max

No clarification needed, if you read the thread, you'll see that I've solved that problem already. Indeed, as you say: "The small wheel mounted on the same hub as the big (outer wheel) slips and drags", I've given a mathematical description that shows that the small wheel must slip if the large wheel rotates without slipping, contrary to the premise that both wheels rotate without slipping. Problem solved.

It seems it's no longer a question of physics and mathematics, but of language and the meaning of sentences. Does "it's essentially the physical principle behind the gearing of a bicycle or motorcycle" mean the same thing as "it is claimed that gears and chains are essential for explaining Aristotle's wheel paradox". In particular, does "essentially" in the first sentence mean the same thing as "essential" in the second sentence? I wouldn't think so, but I suppose there must be many people on this forum with a better knowledge of the English language than I have, perhaps they can enlighten me? Why is everyone here, with a few exceptions, so silent about this matter? Where are all the big names who give their opinion on practically any subject discussed here? Is it embarrassment?

The point of my previous reaction was just that Jonathan's remark about the principle behind the gearing is not relevant (and certainly not "essential") to the paradox and that, contrary to what you said before, he nowhere suggests that it is, it was just a side remark. Perhaps you can start a separate discussion about the principles behind the gearing of a bicycle, but that should not distract from the discussion about Aristotle's wheel paradox.

The only occurrence of the term “essential” in that link is in this sentence: “It's essentially the physical principle behind the gearing of a bicycle or motorcycle..”. That is obviously not the same as saying: “gears and chains are essential" [for explaining the paradox], as you're suggesting. Nowhere does Jonathan say anything like that. What he does say in that sentence, has a completely different meaning, namely that the theory behind the explanation of the paradox is also in essence the principle behind the gearing of a bicycle or motorcycle.

The Wikipedia article states "The wheels roll without slipping". What does this "slipping" then refer to? Slipping with regard to what? The term "slipping" implies the interaction between two different surfaces, in general a wheel and its support, such as a road or a rail. Without implied supports for both wheels, the Wikipedia statement would make no sense. If the article doesn't explicitly mention the terms support or surface, it certainly implies them with the abovementioned statement. Perhaps you don't like the physical description. Then you can translate the problem into mathematical terms: wheels become circles, supports become lines. etc. But there is a one-to-one correspondence between both descriptions. In previous posts I've also shown how the physical process of slipping between wheel and support can be translated into mathematical terms, so I won't repeat that here.

I've added another emphasis: the wheels roll without slipping for a full revolution.The non-slipping is mentioned for both wheels, which makes only sense when each wheel has its own support (tangent). And that premise is also the source of the paradox: the wheels cannot both simultaneously roll without slipping: when the instantaneous bottom point of the larger wheel has a translation speed = 0 (no slipping), the instantaneous bottom point of the smaller wheel necessarily has a translation speed > 0, in other words, it is slipping. And that is all there is to Aristotle's paradox.

Even if you have difficulty visualizing the slipping of the (in this example) smaller wheel, against its support, a mathematical analysis makes it crystal clear that it is in fact slipping if the larger wheel is rotating without slipping against its own support. I don't know how to link to a single post, so I copy here my earlier post: To slip or not to slip is mathematically expressed by the translation speed (with regard to the rest frame of the support) of the instantaneous lowest point on the rolling wheel (circle) where it touches the support (the considered tangent line). If that translation speed is zero, there is no slipping, a non-zero (in this example >0) translation speed implies slipping. Any questions?

It doesn't matter whether you talk about wheels or about circles. One is a physical description and the other one the mathematical equivalent. Translation from one description to the other one is no problem: the rims of the wheels are the circles and the supports (road/rail/etc.) are the lines. The important condition is what in the original article is called "unrolling the line", "tracing out the circumference", and in mechanical terms "rolling, i.e. rotating without slipping". The origin of the paradox is the supposition that both wheels (that form one rigid body with a common center) can rotate without slipping/can trace out their circumference. Suppose the large wheel/circle rolls without slipping. After 1 period in time T the center of the circle is translated over a distance 2*pi*R, with a uniform translation speed of its center v= 2*pi*R/T. The point at the top of the circle is translated with speed 2*v and the bottom (that touches the line (=support) has translation speed zero. The translation speed of the point at the top of the smaller circle ≡ v2 = 2*pi*(R+r)/T. This can be checked by substituting r=R and r=0. Similarly, the translation speed of the point at the bottom of the smaller circle ≡v3= 2*pi*(R-r)/T > 0 for r < R. So we see that for the smaller circle and its tangent (support) the condition for tracing out the circumference is not met. That the bottom point of the smaller circle has a translation speed > 0 is the mathematical equivalent of saying that the smaller wheel is rotating and slipping. So the notion of slipping is essential to the solution of the paradox. If you don’t like the word, you can say it in mathematical terms: it is not possible that the bottom points of both circles during rotation have zero translation speed. But it is just the same as saying that it is not possible that both wheels rotate without slipping.

Here is another graphic, illustrating the solution of Aristoteles' paradox. The two concentric circles represent the corresponding wheels, with respective radius r and radius R, with r = 2/3 * R. The outer wheel rolls without slipping over its support. Two instants are given: the start position and the position after a rotation over 3/4 pi radians. The blue spoke points north in the start position and southeast in the second position, the black spoke points northeast in the start position and south in the second position, etc. If the small wheel would roll alone, without slipping over its support, its position would after a rotation over 3/4 pi radians be given by the grey circle, with its center at r*pi*3/4. But when it is fixed to the outer wheel and the outer wheel rolls without slipping, rotating over pi*3/4 radians, the smaller wheel is carried along by the large wheel and is translated over a distance R*pi*3/4. Its translation by rolling alone, rotating over the same angle, would result in a translation over only 2/3*R*pi*3/4. The difference (R-r)*pi*3/4= R*pi/4 must be made up by slipping of the smaller wheel over its support. The movement of the smaller wheel is thus a combination of rolling and slipping, in this case 2/3 rolling and 1/3 slipping. In the idealized case these proportions are fixed over the whole traject, valid for any time interval, no matter how small. In real life situations there would be tiny fluctuations, but over longer intervals the proportions would be the same.

> Clear, simple, and wrong four ways. There aren't two wheels and two supports. From the Wikipedia reference in your very first post in this thread: "There are two wheels, one within the other, whose rims take the shape of two circles with different diameters. The wheels roll without slipping for a full revolution." The accompanying figure clearly shows that those "two wheels, one within the other" move over their respective supports, so there are two wheels and two supports. > There is one of each for an ordinary wheel. The crux of the paradox is that the inner "wheel" moves farther than its circumference with one full rotation. You mention translation, then abandon it in > favor of "slipping." You should read better, I wrote " [both wheels...have] the same translational velocity (of their common center). I then show that the rim of at least one of the wheels must be slipping against its support. > This video (and many others) explain rolling without slipping (or skidding) andtranslation. What part of “without slipping” do you not understand? This rather elementary video tells nothing that contradicts my text. > Likewise, the smaller "wheel" does not slip nor skid. An inner “wheel” slipping on an imaginary road is as silly as a person slipping on imaginary ice. Aristoteles' example may be imaginary, but it can be easily and unambiguously realized in reality. Of course such realizations show the real slipping, predicted by the mathematical analysis. > Translation fully accounts for its moving the horizontal distance 2πR, like it does for its center and the wheel with radius R and the same center. The video makes that clear. Yes, the movement of the common center can be described as a translation (as I did in my text), but the rim of at least one of the wheels must be slipping when that common center is translated. > This article does not clearly distinguish between slipping and skidding, but it can be done. In essence slipping is rotation without translation, such as a wheel of a car on ice or stuck in > snowdoes and the driver pushes hard on the accelerator pedal. In essence skidding is translation without rotation, such as a wheel of a car does on an icy >road and the driver pushes hard onthe brake pedal. Both are due to a lack of traction and affect the translation movement of the entire wheel uniformly. An inner "wheel" slipping on an >imaginary road is as foolish as a person slipping on imaginary ice. Such foolishness implies translation movement is not uniform – a smaller inner "wheel" "slips" more than a larger inner "wheel." Translation of a rolling wheel (with or without slipping) is of course not uniform. The translation of a point on the rim is for example different from the translation of the center of the wheel. As you can see in the above-mentioned elementary video. [..] > The following proof is simple and correct. > The distance a circle moves translation-wise is always the same distance as its center moves. Since a wheel and any inner circle concentric with it have the same center, the wheel andsaid circle > always move the same distance translation-wise. QED. This is true independent of any rotation, slipping, or skidding. This formulation is rather vague: what is exactly the distance that a circle moves translation-wise? Different points on the circle/rim of the wheel move in an arbitrary time interval different distances translation-wise. But for two concentric rotating circles/wheels the distance the centers move is the same, as they have the same center.*) And that is exactly part of the argument I've used. It is only one half of the proof. In itself it doesn't tell us anything about the paradox. The second part of the proof is the realization that the paradox is created by the supposition that both wheels can rotate without slipping against their respective supports (see the Wikipedia article and the accompanying illustration). That is not possible, if one wheel rotates without slipping, the other one must be slipping (/skidding if you like, I make no distinction), not only in theory, but also in reality. If you realize such a system, the slipping is unavoidable and very real, it‘s not some kind of illusion, on the contrary, it is an essential part of solving the paradox. *) In fact, this is part of description of the paradox (in the Wikipedia article: "the two lines have the same length"). The paradox gives an argument that these lines must have the same length, and another argument that these lines have different lengths. That is a contradiction, and therefore we call this a paradox, but without inspecting both arguments, we don't know how this paradox can be solved. As I've said many times, the error is in the statement (Wikipedia:) "The wheels roll without slipping for a full revolution". From the other argument follows that this is impossible, both wheels cannot roll without slipping, at most one wheel can, and the other wheel must therefore roll with slipping. This non-slipping vs. slipping is not just some side-issue, it is the origin of the existence of this paradox. By identifying it, we solve the paradox.

The crux of the paradox is the implied -and false- suggestion that both wheels can turn without slipping on their respective supports (rail or road etc.). Both (concentric) wheels are part of a rigid body, so they have the same rotational velocity and the same translational velocity (of their common center). When the larger wheel makes one rotation without slipping, it travels over a distance of 2 π R. So does the smaller wheel, but if this wheel wouldn't slip, it would only travel over a distance 2 π r (r < R). However, it has to travel over a distance of 2 π R, so apart from its rotation it must also slip with respect to its support, to keep up with the larger wheel. Mutatis mutandis if it is the smaller wheel that rotates without slipping. It's all so very simple and trivial, so why should we have a discussion that now covers already 25 pages? It isn’t that difficult!

What a nonsense. According to these criteria astronomy wouldn't be an objective science either. The same can be said of the stars, astronomers can only observe some photons arriving on earth. We can’t directly observe the evolution of a star, so the theories about such stellar evolution aren’t objective science either?

I think that's just a matter of definition. As I said before: a paradox is an argument that leads to an apparent contradiction. The contradiction doesn't exist in reality, so there must be some error in the argument. One can solve the paradox by showing were the error in the argument lies. The bad argument, i.e. the paradox, doesn't disappear in my opinion, it has only been shown what was wrong in the argument Well-known paradoxes are for example those special-relativity paradoxes, such as the barn-pole paradox, which seem to imply contradictions in reality, by incorrectly supposing that simultaneity is an invariant. There is no contradiction, but the paradox does exist and is well-known. I think the quibble is about the distinction between a paradox and a contradiction.

Perhaps you should read this first: https://tinyurl.com/yas5draz

That’s also an excellent video. It demonstrates clearly the solution of the Aristoteles paradox.

The cycloids are not relevant for the solution of the paradox, as they are a description of the movement of one point of the wheel in the z-x plane (z = up, x = direction of rail/ledge/road). The paradox is about the interface wheel-rail/ledge/road, however. That is: the points of the wheel and of the rail/ledge/road where they touch each other. The position of these points form a straight line along the rail/ledge/road. When the wheel rotates without slipping on its support, the length of that line is 2πR after one revolution of the wheel with radius R. With the two concentric wheels (radius R and r, r < R) in the paradox, the length of those lines would after one revolution without slipping be equal to 2πR and 2πr respectively. However, the actual length can only have one value, as those wheels are part of a rigid body, so at most only one wheel can rotate without slipping, for example the larger, outer wheel. The smaller inner wheel then has to travel the same distance 2πR over its support. In the same time interval its proper slip-free rotation distance is only 2πr, which is not enough, so it has also to slip over a distance 2π(R – r) to keep up with the outer wheel. Jonathan’s animation shows this clearly. Further, I’m reminded of this joke: https://tinyurl.com/y7hly2al

It's interesting to note that many commenters on that video give the correct and simple solution of the paradox.

I can't make head nor tail of this. You suggest that the slipping on Jonathan’s animation is exaggerated. Does that mean that you accept just a little bit of slipping, as long as it isn’t too much? In fact it is really easy to see how much the inner wheel/protrusion is slipping if we assume that the outer wheel/the rim is rolling without slipping: for one revolution it’s just the difference between the circumference of the outer wheel and the circumference of the inner wheel. It’s easy as that, no need for cycloids to solve the paradox, and it’s perfectly illustrated by Jonathan’s video.

Very nice demonstration!

24 minutes ago, merjet said: Apparently Max failed to grasp the first sentence of the Wikipedia page: "This article's factual accuracy is disputed." Oh, I thought this was the reference you gave for your original question. I can't remember that you said that the problem was there incorrectly stated, but I'll admit that I didn't read the whole thread. That the factual accuracy is disputed doesn't necessarily mean that it is incorrect. I read the talk page, but I wasn't impressed by the quality of the comments there. But if the description is not correct, perhaps you can give us then the correct version of Aristotle's wheel paradox?

Indeed, the clue to the paradox is in this sentence of the Wikipedia article: "The wheels roll without slipping for a full revolution". It is impossible however that both wheels roll without slipping, as the wheels have different diameters, so when the path of one wheel equals its circumference, the path of the other wheel cannot equal its circumference. Therefore, when one wheel rolls without slipping, the other wheel must be slipping with respect to its rail (or the road). So the path of at most one wheel can have the length of its circumference. You don't need more to solve the paradox. Whether there exists a paradox is rather a matter of semantics. In general, a paradox is an argument that leads to an apparent contradiction, for example by using a fallacious argument or starting from a false premise. In reality there is no contradiction, in this case while the premise that both wheels can simultaneously roll without slipping is false. Solving the paradox is just showing what the error in the argument is. There is no contradiction, and the paradox is just a bad argument.

Holy Jesus... I started to read this thread, but gave up somewhere halfway, as the solution is so trivial. Jonathan c.s. are of course right: the origin of the paradox lies in the supposition that *both* wheels are moving without slipping. That is of course impossible: if the wheels are part of a rigid body, they rotate with the same angular velocity. After one revolution the small wheel travels a smaller distance than the large wheel (2 pi r vs. 2 pi R; r < R). If the outer wheel rotates without slipping and travels a distance of 2 pi R, the inner wheel *must* also travel 2 pi R during one revolution (it's a rigid body). However, its own translational movement due to rotation is only 2 pi r, so it *must* be slipping to make up the difference. Of course we could also suppose that it is the inner wheel that moves without slipping, in that case the outer wheel must slip (moving slower than its "natural" movement), and finally it's also possible that both wheels are slipping (one going faster, the other going slower than their "natural"movement). What will happen in reality depends on other mechanical conditions, such as the magnitude of the respective friction forces.

I hear no illusion at all, I just hear a repeating series of ascending "glissando" tones. I can clearly perceive each time the onset of a new bass tone of the next glissando. Sounds like something written by Czerny...

That should be: "de gustibus non disputandum est".