Coin rotation paradox #2

Four months ago I edited the Wikipedia article about the coin rotation paradox (link), adding the new section Analysis and Solution. I explained why the moving coin does what it does. I wrote about it on this blog here May 29.

More recently somebody else (user name Cmglee) added another section about one coin revolving around another, but the two coins having unequal radii. While the person gives the correct formula, why it is correct is not explained. The person also gave a reference, but the reference neither gives the formula nor explains why it is true for any two radii. Therefore, I added a new section to the Wikipedia Talk page as follows.

The formula R/r + 1 is correct. However, the reference – author Y. Nishiyami – doesn't give the formula, much less prove it is true for any R and r. Also, Nishiyama says, “Separating revolution from rotation is helpful for understanding, but doing so does not provide a fundamental solution.” This latter part is not true. For a coin with radius r to make one revolution around a stationary coin with radius R, the center of the moving coin travels a circular path with radius R + r. (R + r)/r = R/r + 1. Likewise, the circumference of the path is 2*pi*(R + r)/(2*pi*r) = R/r + 1 times the circumference of the revolving coin. Rotation is irrelevant. [End]

The key to this solution is to focus on the center of the moving coin. The center travels a circular path with radius R + r and hence circumference 2*pi*(R+r) in order to make one full revolution around the stationary coin. How many rotations it makes, even zero, while doing so is irrelevant. (Rotation versus revolution.) If it "rolls without slipping or sliding," then the number of rotations will also be R/r + 1.