Max

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    Max Keller

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  1. 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.
  2. 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.
  3. 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?
  4. Max

    Aristotle's wheel paradox

    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.
  5. Max

    Aristotle's wheel paradox

    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.
  6. Max

    Aristotle's wheel paradox

    > 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.
  7. Max

    Aristotle's wheel 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!
  8. 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?
  9. Max

    Aristotle's wheel paradox

    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.
  10. Max

    Aristotle's wheel paradox

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

    Aristotle's wheel paradox

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

    Aristotle's wheel 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
  13. Max

    Aristotle's wheel paradox

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

    Aristotle's wheel 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.