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Problem of the Week 813

Blurred Spokes

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Here's a great new problem!

In a photograph of a rolling bicycle wheel, which parts of which spokes will be least blurred?

-- From work of R. de Souza and G. L. Vasconcelos,
Universidade de Pernambuco, Recife, Brazil
(to appear in American Journal of Physics, 1996)

We assume that the wheel has spokes connecting the round rim to the exact center. The wheel is assumed to roll in a plane parallel to the plane of the film; the camera does not pan. The shutter speed is assumed to be slow enough so that there is some blurring, but fast enough so that some parts of some spokes are in focus. The problem is to determine which parts of which spokes will be most perfectly focused.

We note that the statement appearing in one physics text - "The spokes near the top of the wheel are more blurred than those near the bottom of the wheel because they are moving faster." - is incorrect. I wish I had time to go out and make a nice black and white photograph of this phenomenon. Photos can be found in physics texts, but it would be nice to have the real thing. Of course, in a real bike the spokes are tangent to a small round hub. This variation can be handled (John Duncan, University of Arkansas, has done so).

Murray Klamkin, problem editor of the MAA magazine, "Horizons", writes: "The best problems are elegant in statement, elegant in result, and elegant in solution. Such problems are not easy to come by." It is inspiring that mathematics continues to provide us with new problems that do fit Murray's criteria. I suggest that this bicycle wheel problem does so admirably!

© Copyright 1996 Stan Wagon. Reproduced with permission.

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2 October 1998