# European Journal of Physics: On the stability of a bicycle on rollers

Unlike all the previous examples discussed, which are self-published and answerable to no one, the European Journal of Physics describes itself as a peer-review journal, and as such, it should be held to a higher standard. That makes the 2011 article “On the stability of a bicycle on rollers” by Cleary and Mohazzabi all the more troubling.

Jim Papadopoulos and I, with the help of Andy Ruina, responded with “Comment on ‘On the stability of a bicycle on rollers’“, but it was treated as though it were merely the other side of some controversial issue with no clearly correct answer. Instead of trying to paraphrase what we spent weeks polishing to perfection, let me just quote directly from our response:

The two key differences noted in the paper, however, are not correct. The first is the authors’ claim (page 1297) that ‘riding a bicycle on rollers explicitly tests the role of the centrifugal force’, repeated in other words as ‘if the bicycle and the rider both lean to one side, there is no centrifugal force to correct the lean on rollers . . . ’. This claim is most easily refuted for treadmill riding, to which the authors suggest their ideas also apply. Riding a bicycle on a constant-speed treadmill is actually (neglecting wind resistance) mechanically identical to riding on fixed, level ground at a constant speed. The most straightforward explanation for this is Galilean invariance, which says that the laws of mechanics are the same in all reference frames moving at a constant velocity relative to each other. To argue against this invariance, one would have to show that the mechanics have changed at some point in the following short series of experiments.

(1) Ride a bike on the deck of a long stationary ship straight toward the stern at 18 km/h, as indicated by a speedometer on the bike’s handlebars.

(2) Ride on the same ship that is now traveling forward at 18 km/h relative to the shore, straight toward the stern at 18 km/h, and thus remain stationary relative to an observer standing on shore.

(3) Bring the ship back to rest, and ride straight toward the stern at 18 km/h on a deckmounted treadmill whose belt is moving straight toward the bow at 18 km/h, and thus also remain stationary relative to an observer standing on shore.

In a video of just the bike and the surface on which it rides, the scene in experiment (3) cannot be distinguished from the scene in experiment (2) because the mechanics are in fact identical. And those, in turn, are identical to the situation in experiment (1). Thus, riding on a treadmill is mechanically no different from riding on fixed ground, and any differences between riding on rollers and riding on fixed ground are, in fact, differences between riding on rollers and riding on a constant-speed treadmill.

The second issue is the paper’s reinforcement of the common misconception that inertia or forward momentum somehow provides stability. This is indicated by the phrases ‘forward inertia . . . aids in bicycle handling for stability’ on page 1293 and ‘the loss of inertia limits the degrees of freedom in bicycle stability’ on page 1299. This point is again simply refuted by the Galilean invariance: in a frame of reference that is stationary with respect to the ground, a bicycle on a treadmill has no forward momentum, yet it has the same balance dynamics (as argued above).

We could not understand Cleary and Mohazzabi’s reply and so discontinued the discussion.