I remember seeing a math problem, in more than one place over the years, about a rolling wheel. You are given the radius of the wheel and asked to determine how far the hub of the wheel moves forward in one rotation of the wheel. My memory of the question must be wrong because that’s a straightforward question. The trick was that the calculation required to solve the problem was counterintuitive, and the answer was not simply a distance equal to the circumference of the wheel.

Maybe it involved a board rolling across the top of the wheel, or some other complication, rather than simply the distance the hub moved.

It’s the kind of thing that would show up in a book of math brain teasers for kids. Ring a bell?

I am going to go out on a limb here, and say that the answer is clearly one. The moving wheel is traveling a distance equal to its own circumference (as in the OP’s “straightforward” question), and the fact that is following a curved rather than a circular path is irrelevant to how many revolutions it makes upon its own axis.

Of course, I am assuming that “fixed wheel” really is fixed, and not free to rotate. I guess that if it rotates too, the answer might indeed be two. But that is an ambiguity in the question, not a math issue.

Try it with two quarters (or other coin of your choice) and you will see that the answer is two. The fact that it’s following a circular path is relevant.

ETA: Here’s a thought experiment: Roll the wheel along a straight line. It will make one rotation (not revolution). Once it reaches the end of the line, keeping the point on the wheel in contact with the same point on the line, curl the line around so that endpoint joins the starting point forming a circle. As you take the wheel along with you to form the circle, the wheel will make another rotation. One rotation because it rolled, another rotation due to the fact that it is revolving around a circle.

The same basic reasoning shows that a sidereal day is shorter than a solar day by just the amount so that there is one more sidereal day in a year than solar days.

Nuh uh. There is a second revolution alright, an orbit around the center of the fixed wheel, but the moving wheel only makes one revolution about its own axis.

I do not understand your thought experiment at all. Curling the straight line around the wheel that has moved will not make it rotate again.

Try my thought experiment. Straighten out the rim of the fixed wheel into a straight line (equal to the circumference of the moving wheel) and roll the other wheel along it. One revolution, as in your original example.

Or try your experiment with the quarters. Start with the one that is to move at the top, and mark where they touch. Now roll it round to the bottom, half way round the fixed one. Your moving quarter will now be the same way up as it was when it started, which may deceive you into thinking it has made a full rotation about its axis. But it has not. The point you marked is no longer touching the rim of the fixed quarter, but is in fact diametrically opposite to the point now touching. Your moving coin has made half a rotation, but the path it was rolling along has also made a half a rotation, in the sense that, where it originally pointed up, it now points down, and this brings the moving quarter back upright again. Nevertheless, to complete a full rotation of the moving quarter around its axis, you will need to roll it the rest of the way round, back to its starting point, thus bringing the marked point back into contact with the rim of the fixed quarter once again.

This is such a simple problem that I wonder why people are talking about “thought” experiments, when it’s trivial to do the “real” experiment:

Lab Report:
The experiment was performed using 2 US dimes, both manufactured by the Philadelphia Mint, one manufactured in 1988, the other in 1993. The “reverse” (i.e. “tails”) of both dimes were oriented facing the observer, with the “torch” icon on each dime lined up in a vertical and upright direction. The edges of the dimes were brought into contact, maintaining this orientation. The upper dime was held with the observer’s right hand, while the lower dime was held by the left hand. The dime held by the right hand was then carefully moved in a counterclockwise direction around the circumference of the dime held by the left hand, maintaining “point-to-point” contact during the entire experiment, while allowing the “right-hand” dime to rotate around it’s axis. The “torch” icon on the “right-hand” dime was then observed as an indicator of rotation around the dime’s axis. The “left-hand” dime was also observed, again using it’s “torch” icon, to ensure that the “left-hand” dime did not rotate around it’s axis. The “right-hand” dime was observed, using the “torch” icon as the reference, to rotate around it’s axis 180 degrees (i.e. become inverted), after traveling 90 degrees around the rim of the “left-hand” dime. As the experiment continued, the “right-hand” dime was observed to continue to rotate around it’s axis, until after traveling a further 180 degrees around the circumference of “left-hand” dime, it was observed to have returned to it’s “torch” icon upright condition. At this point, it was observed that the “right-hand” dime, which was in the “upper” position, when the experiment was begun, had transitioned to the “lower” position, relative to the 2 dimes original position. This was counted as 1 rotation about the “right-hand” dime’s axis. The experiment was continued in the same fashion until the “right-hand” dime returned to the same orientation relative to the “left-hand” dime (i.e. “upper” vs. “lower”), that the experiment had been begun in. Again, the “right-hand” dime was observed to have gone through the same orientation sequence as had been observed during the previous portion of the experiment. Again, the “right-hand” dime had been observed to have rotated through the same sequence as was counted as “1 rotation” previously in this experiment. This was also counted as 1 rotation, leading to a total “rotation-count” of 2. The researcher concludes, therefore, that given the experimental problem, that when the experiment is performed, that 2 rotations of the coin in question will result.

The answer is 2, and when you do the “real” experiment, it’s obvious why it takes 2, rather than 1, even if it’s hard to state in words. As a bonus question, there is a major error in the above lab report. I didn’t feel like editing to correct it.

It’s two, like everyone says. The same principle applies to regular polygons with any number of sides: Swing an equilateral triangle around another equally sized equilateral triangle, corner by corner, and what happens? The triangle turns 240 degrees at each corner, for a total of 3 * 240 degrees = two revolutions. Swing a square around another equally sized square corner by corner, and what happens? The square turns around 180 degrees at each corner, for a total of 4 * 180 degrees = two revolutions. Etc., etc. And a circle is just like a regular polygon with a huge number of sides.

As others have noted, the correct answer is 2 rotations.

The path described by the center of the moving wheel is a circle with radius of 2r, where r is the diameter of the two identical wheels. The distance traveled by the center of the moving wheel is thus 2 times pi times the radius, or 4 times pi times r. Since the distance traveled by the center of the moving wheel is related to the angular rotation (measured in radians) by theta times the radius (of just one wheel), we have theta times r = 4 times pi times r, so theta = 4 times pi, or two rotations.

Everyone who answered one rotation owes the rest of us a treadmill.

Small correction: the bolded word should be “radius”, of course. But that is a very nice proof. I was about to post an alternative proof for the polygonal case based on calculations of the relevant angles, but your proof is so much nicer, there’s no longer any point.

Your interpretation only makes sense in a rotating reference frame, but that’s not a sensible interpretation of the question. I suppose you don’t regard the moon as spinning around its axis either?

Gold star for you, MikeF. Go to the head of the class, stand in front of your classmates, stick out your tongue and go “Nyah, Nyah”. You deserve it, you “science genius”!

Unless you count FDR on the obverse (i.e. “heads”) as an “animal”, then there are no animals on US dimes of that design. So: No animals were mistreated in this experiment. I did, however just go spend the two dimes used in the experiment to buy some booze, and after drinking it, I may go kick one of my cats, so it’s still an open question whether any animals will be mistreated in conjunction with this experiment…

It took me a very long time to figure out why this was true, because I kept doing it in three dimensions, with one wheel perpendicular to the other. I kept coming to the conclusion that it didn’t matter what the angle between them was, it was effectively the same as if they were just sitting on top of each other. But when you align them anti-parallel, you introduce a second rotation that in some sense was always there, but hidden by the fact that if you have them at any other angle, the rotations are not in the same plane.