The wheel paradox that stumped Aristotle and Galileo

Here's a simple problem, illustrated simply, that will have you cocking your head and wondering how it's done. You won't be the first. Aristotle (reputedly) first took a whack at this, and Galileo gave it a try as well. See what you can make of it.

Not everyone agrees that Aristotle invented this little paradox, but everyone agrees that it would be just like him to come up with something like this. The paradox involves two different-sized wheels, one inside another. Think of the edge of your tire and the edge of the hubcap. The two rotate in sync, and they rotate over a certain distance. But should they rotate over the same distance?

The wheel paradox that stumped Aristotle and Galileo

If you look at the animated gif above, both wheels use their entire circumference to trace the same amount of distance - the red line. Clearly one circumference is smaller than the other. Either that means that the wheels have the same circumference, which they don't, or that different circumferences "unroll" to the same length, which they can't. (If they did, since this is true no matter how small the radius of the wheel, technically a wheel with the circumference of an inch should be able to go the same distance in one roll as a wheel with the circumference of a mile. The only thing that's keeping us from being able to drive across the country with one revolution of our tires, then, is that the tires aren't small enough.)

That can't possibly be right. The smaller radius can't possibly be equal to the larger one, so what's going on? An easy answer to this is to trace, not the line, but the journey that each segment goes on to get from one point of the line to the other. Go ahead and take your finger and move it with the line showing the radius of the circle, tracing the arc that the smaller circle goes through to get from one point to the other. Now trace the arc that the larger circle goes through to get from one point to the other. It should be obvious that a point on the larger circle goes though a larger arc, and therefore a longer journey, to get to the same point.

And what's happening on the red line? To answer that, we'll invoke the wheel-and-hubcap image again. If you have parked badly, with your wheel on the street and your hubcap on the curb, exactly what do you expect to hear and feel when you pull away? If the answer is the smooth glide of a tire gripping the road, and the hideous skreech of metal slipping over concrete, you're right. The inner wheel, when made to trace out the same line as its larger compatriot, will slip. They don't make the same journey.

Top Image: dbking

[Via Mechanics, Wolfram Math World.]