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© Copyright 2002, Jim Loy
We have two identical coins. And we roll
the one on the left halfway around the other coin, so it rotates without
slipping against the other coin, so that it ends up on the right of the other
coin. It has rolled over a length of only half its circumference, and yet it
has made one complete rotation. If it started right side up, then it ended
right side up. Does that make sense? Think about it.
Answer: Of course it makes sense, and is no
paradox, despite the title of this article. But it surprises people, and is a
famous puzzle. On the right, we see one source of the surprise. The coin rolls
a distance C (its circumference) along a straight line, and ends right side up.
After a distance C/2, it is upside down. Obviously the situations are
different.
Just to give you a little more experience with rolling
coins, look at the diagram on the left. Here a coin "rolls" around a
point. It goes zero distance (approximately) and makes one half rotation. So
the distance rolled along a curved path has little to do with the coin's
rotation, as viewed by a non-rotating viewer, like yourself.
There are different ways to look at the first diagram, the original "paradox." From a viewer on the stationary coin, the rolling coin did make only one half rotation. Or we can add rotations; the coin made 1/2 rotation + it made a U-turn around the other coin, for a second 1/2 rotation (like our third diagram). The sum is one complete rotation. Still another way of looking at it is that the center of the moving coin is not moving a distance of C/2, but C. So, even though the point of contact has moved a distance of C/2, the coin has move a distance of C. And it should rotate once.
Anyway, it may have caught you sleeping. But it is not a paradox.