Reply to ‘Comment on “On the stability of a bicycle on rollers”’

This communication is a Reply to a Comment on our article ‘On the stability of a bicycle on rollers’. We are glad that our paper Cleary and Mohazzati (2012 Eur. J. Phys. 32 1293) has garnered such vigorous debate and interest. Our paper addresses the stability of a bicycle on rollers as opposed to that on the ground. The authors of the Comment (Dressel and Papadopoulos 2012 Eur. J. Phys. 33 L21) suggest that centrifugal forces are present on rollers by using a treadmill as an example. A significant portion of our paper is about the stability of bicycles on rollers, with a caveat on how riding on a treadmill would be different from that on rollers (p 1299). We highlight some conveyor belt examples to discuss certain stability processes, but in the paper do not clearly make the distinction that we consider the rollers to be a more unique case than a conveyor belt and this may have led the authors of the Comment to assume that we assert that all the same stability factors apply in both cases. We agree with the authors of the Comment that a treadmill operated at a constant velocity does constitute an inertial coordinate system and that Galilean invariance applies; therefore, a bicycle can experience centrifugal force. We also agree that the rider on rollers must stabilize the bicycle within a narrow range, the equivalent of which would be like riding atop a brick wall or along a narrow path. We address this in the paper in reference to a restricted angle to steer (p 1298), compensation for the straight path (p 1298) and requirement that the bicycle stays perpendicular to the rollers to stay on them (p 1298). Rollers, on the other hand, do not constitute an inertial reference frame, simply because the rotation of the rollers is coupled to and dictated by the rotation of the bicycle wheels. If the rollers are driven at a constant speed by a motor, or if the rider pedals at a constant rate so that the rollers again turn at a constant speed, of course one can view them as constituting an inertial reference frame. However, the rider does not always pedal at a constant speed and any change in the rate of pedalling will change the speed of the rollers, hence making them, at least in principle, a non-inertial reference frame. Also any change in the direction of the wheel with respect to the roller can change the rate of spin of the roller. Therefore, strictly speaking, Galilean invariance does not apply and rollers cannot be viewed as similar to a ‘ship 0143-0807/12/040025+02$33.00 c © 2012 IOP Publishing Ltd Printed in the UK & the USA L25 L26 Letters and Comments travelling forward’ at a constant velocity and a bicycle riding on that ship in the opposite direction at the same speed. This is particularly relevant when discussing centrifugal forces because it confounds the role of coupling the roller speed to wheel speed with the lateral stresses attributed to the centrifugal force. A more apt example would be standing on a drifting canoe or paddle-board and walking from the bow to the stern of such a vessel. In this example, the frame of reference of the canoe or paddle-board would not be inertial because its velocity relative to the ground is influenced by the movement of the person walking along it which is similar to a rider on rollers. A test of this assertion would be to study a bicycle on rollers that are powered by an external motor at constant speed, which would constitute an inertial reference frame. Nonetheless, due to friction between the tyres and the rollers, there can exist a force perpendicular to the bicycle wheels, which the authors of the Comment refer to as a lateral force. This can be seen in equation (5) on p 1298 of our paper. These forces come into play when the front wheel of the bicycle turns, or the centre of mass of the rider and bicycle leans to the side. This allows the bicycle to not slip from beneath the rider. Even though the frame of reference of the rollers does not qualify as an inertial frame, these forces are still centrifugal in nature. On a curved path, this would be considered a stable geometry for the rider. However, on rollers these forces are very small and do not aid in bringing the centre of mass of the system out of a lean, which is an unstable geometry for the rider on rollers. Because of the difficulty to take advantage of the centrifugal or perpendicular forces on rollers, a rider does not use them to correct for a lean and therefore does not depend on them in the same way as when in motion with respect to the ground. Riding on rollers trains a cyclist to not use these forces for stability. As a result, these forces effectively play very little role in the dynamic stability of a rider on a bicycle in that specific situation. Riding a bicycle on rollers feels closer to but not exactly like riding on ice as opposed to a paved surface. To test the magnitude of these small lateral forces of a bicycle on rollers, an experiment could be conducted to measure the lateral stresses on the roller axle. Finally, we would like to thank the authors of the Comment for bringing these issues up for discussion.