Arms race escalates

By Burkard Polster and Marty Ross

The Age, 15 September 2008

Take a close look at Australian wheelchair champion, Kurt Fearnley. His arms and shoulders are impressive, but what’s surprising is they are equally impressive.

Kurt spends a lot of time racing around a track in the counterclockwise direction. During this time, the right wheel of his wheelchair is further out, so it travels further in each lap. But this should mean Kurt’s right arm is working harder than his left: so why isn’t Kurt’s right arm built up more, just as we see with right-handed tennis players?

A little simple mathematics explains it all. To see what is going on, let us say that Kurt’s wheelchair is 1 meter wide. Then let us imagine Kurt racing around the simple track pictured, an equilateral triangle of the same width of 1 meter. So, Kurt’s left wheel follows the inner edge of the track, and Kurt’s right wheel follows the outer edge.

The diagram makes clear that the extra distance traveled by the right wheel is just the three blue arcs of 120 degrees. These arcs together exactly form a circle with radius equal to 1 meter. So, the distance we are looking for is the circumference of this circle: using the tried and true formula C=2πR, we arrive at an extra distance of 6.28 meters.

The thing to note is that this extra distance is the same, no matter the size of the triangle. And something even more surprising is true. With a little calculus we can show that the same 6.28 meters arises for any shape loopy racetrack.

Now, if Kurt races for 10000 meters on a 400 meter track, then he will complete 25 laps. So, his right wheel will travel about an extra 25 x 6.28 = 157 meters. That’s quite a bit, but definitely not enough to make his right arm bulge more than his bulging left arm.

Our ponderings on Kurt’s wheelchair are closely related to a famous mathematical puzzle. Imagine a loop of that fits snugly around the Equator of the Earth. Now lengthen this loop by one meter and imagine the new loop hovering above the ground. How far is the loop above the ground? What if you consider the same question for a rope around a basketball? As always, contact us if you want a hint.