The Bicycle Paradox
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The Bicycle Paradox STAN WAGON
M
ark Levi’s recent book review [3] in this journal reports on an error made by V. I. Arnold. But the exact nature of the error needs clarification. Here is what Arnold wrote:
What Force Drives a Bicycle Forward? The lower pedal of a bicycle standing still on a horizontal floor is pulled back. Which way does the bicycle go, and in what direction does the pulled back lower pedal move with respect to the floor? Arnold’s diagram is reproduced in Figure 1. Here x is the (infinitesimal) distance traveled left by the pedal relative to the bicycle. His conclusion is that for a particular choice of parameters, the bike (z) moves forward a distance 5x with respect to the ground and the pedal therefore moves forward 5x x ¼ 4x. Arnold’s error is subtle. He is correct that if x moves left relative to the bike (i.e., the pedal rotates clockwise), then the bike moves forward by 5x. But his error is in the assumption that the pedal rotates clockwise when pulled left. The opposite is true! This counterintuitive fact is known as the bicycle paradox. Readers unfamiliar with this surprising behavior are encouraged to find a bicycle and try it. The book editor thought that Arnold was thinking of a cyclist sitting on the saddle in the standard position and pushing the pedals. If so, all that Arnold says is correct. But this is surely false (as Levi noted); the normal riding of a bike uses shoes and friction to propel one forward, and there is no mystery to resolve. Arnold simply was unaware of the bicycle paradox. His mathematics is correct, but he made a physics error (Figure 2). In his review, Levi wrote that ‘‘Arnold’s answer is correct in reference to the bike itself rather than the pedal.’’ This comment is false for any standard bike, though it is correct if the bike has an exceptionally low gear ratio. Here we present the geometry and physics that provide a simple and complete explanation for the various behaviors. The video [1] by George Hart is excellent. In addition to performing the experiment on an actual bicycle, he also shows how the result can be different for a specially constructed bicycle with a very low gear ratio. Throughout this note we assume a 1:1 gear ratio, but the methods apply to any gear ratio.
Arnold’s use of ‘‘pulled back’’ disguises the fact that there are two ways to do it. One can literally move the pedal back, using one’s hand or a string. Or one can apply a force to the pedal (such as the impulse of a hammer, or a strong wind focused only on the pedal). The force interpretation is more general, and a solution has to show first that the force causes the pedal to move backward relative to the ground; then one has to show that this backward motion yields counterclockwise rotation. The laws of physics imply that a leftward force causes leftward motion of the pedal. Using a weight on a wire that pulls the pedal left by means of a pulley, one can arrange that the force in question is gravity. If the force caused rightward motion of the pedal, then that would raise the wei
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