# Max

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Max Keller

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.

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!
5. ## improving arguments Annotating Regi Firehammer's essay on Evolution

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?
6. ## Einstein at the blackboard

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.