This question is inspired by this YouTube video which asks a question about how many times a circle revolves when rolled around another circle and comes up with a counterintuitive answer. I have a related question:
Suppose we have two (perfectly equal in size) quarters. We pick one to be stationary and call it $B$. We pick the other $A$ to be in motion. We roll $A$ around $B$, how many revolutions does $A$ revolve?
I approached this problem by noticing that the center of $A$ will stay a constant distance around the center of $B$. If $r$ represents the radius of each of the coins, that distance is $2r$. The center of $A$ will travel in a circular path fixed about radius $2r$.
The distance travelled by the center is $4\pi r$. In order the revolve once, the center of circle $A$ must travel a distance equal to its own circumference, which is $2\pi r$. Taking the ratio of the two we get that the moving coin should have made $2$ revolutions.
Is this a valid approach to the problem?