Aristotle's wheel paradox #4

Wikipedia has a page for Aristotle's Wheel Paradox. I edited it substantially during September-December 2018. The page was very poorly written before then. I added two original solutions, the second and third below. The page has been edited many times since then, but only in minor ways. On August 18, 2021 somebody else edited the page, greatly reducing in size two images I had put on the page and putting them into frames on the right side of the page. Wikipedia allows an editor to preview how his or her edits will affect the page's appearance on the device he or she is using. However, the page's appearance on my smart phone -- and likely most or all other smart phones -- is quite different due to its small screen size. For example, the frames do not appear. Their content still shows, but the article's flow and appearance are worse.  

Most Wikipedia pages can be edited by millions of people whenever they get the urge. So I am hereby preserving part of the article as it existed before August 18 in this blog, where presumably nobody but me can modify it. I will not so preserve the section titled History of the paradox, which I did not edit. The following are the introduction and the Analysis and solutions section, which were written near 100% by me. 

Introduction

Aristotle's wheel paradox is a paradox or problem appearing in the Ancient Greek work Mechanica traditionally attributed to Aristotle. A wheel can be depicted in two dimensions using two circles. The larger circle is tangent to a horizontal surface (e.g. a road) that it can roll on. The smaller circle has the same center and is rigidly affixed to the larger one. The smaller circle could depict the bead of a tire, a rim the tire is mounted on, an axle, etc. Assume the larger circle rolls without slipping (or skidding) for a full revolution. The distances moved by both circles are the same length, as depicted by the blue and red dashed lines and the distance between the two black vertical lines. The distance for the larger circle equals its circumference, but the distance for the smaller circle is longer than its circumference: a paradox or problem.

The paradox is not limited to a wheel. Other things depicted in two dimensions show the same behavior. A roll of tape does. A typical round bottle or jar rolled on its side does; the smaller circle depicting the mouth or neck of the bottle or jar.

There are a few things that would be depicted with the brown horizontal line in the image tangent to the smaller circle rather than the larger one. Examples are a typical train wheel, which has a flange, or a barbell straddling a bench. In this case the the distances moved by both circles with one revolution would equal the circumference of smaller inner circle. A similar but not identical analysis would apply.

Analysis and solutions

First solution

The paradox is that the smaller inner circle moves 2πR, the circumference of the larger outer circle with radius R, rather than its own circumference. If the inner circle were rolled separately, it would move 2πr, its own circumference with radius r. The inner circle is not separate but rigidly connected to the larger. So 2πr is a red herring. The inner circle's center is relevant, its radius is relevant, but its circumference is not.

Second solution

This solution considers the transition from starting to ending positions. Let Pb be a point on the bigger circle and Ps be a point on the smaller circle, both on the same radius. For convenience, assume they are both directly below the center, analogous to both hands of a clock pointing towards six. Pb travels a cycloid path and Ps a curtate cycloid path as they roll together one revolution.

While each travels 2πR horizontally from start to end, Ps's cycloid path is shorter and more efficient than Pb's. Pb travels farther above and farther below the center's path – the only straight one – than does Ps. The image below shows the circles before and after rolling one revolution. It shows the motions of the center, Pb, and Ps, with Pb and Ps starting and ending at the top of their circles. The green dash line is the center's motion. The blue dash curve shows Pb's motion. The red dash curve shows Ps's motion. Ps's path is clearly shorter than Pb's. The closer Ps is to the center, the shorter, more direct, and closer to the green line its path is.

If Pb and Ps were anywhere else on their respective circles, the curved paths would be the same length. 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.

Third solution

This solution only compares the starting and ending positions. The larger circle and the smaller circle have the same center. If said center is moved, both circles move the same distance, which is a necessary property of translation and equals 2πR in the experiment. Also, every other point on both circles has the same position relative to the center before and after rolling one revolution (or any other integer count of revolutions). For a wheel with multiple concentric inner circles, each circle's translation movement is identical because all have the identical center. This further proves the circumference of any inner circle is entirely irrelevant (when the outer, larger circle is the one that rolls on a surface).

Aristotle's wheel paradox #1  Feb 18, 2018

Aristotle's wheel paradox #2  Feb 22, 2018

Aristotle's Wheel Paradox #3 Nov 10, 2018