I recently wondered whether there is a simple proof that circular wheels maximise mechanical efficiency. By this I mean:
Show that for a wheel with a given width and cross-sectional area, the circular wheel reaches the bottom of an incline faster than a wheel with any other shape.
This sounds like a rather elementary problem in classical mechanics but I haven't found a proof of this fact in any of the classical mechanics texts that I have read.
Note 1: I assume that the distribution of mass is uniform across the wheel so comparable wheels will have equal mass.
Note 2: Dry friction is assumed to be present.
Answer
A complication arises as the shape gets further from circular symmetry. When it gains sufficient angular velocity, any non-circular object can 'jump' off the incline. See A jumping cylinder on an inclined plane. On leaving the plane the object becomes a projectile. Whether this speeds the descent or not might be difficult to determine.
Another complication is that the circle is the only shape which has neutral stability. It will roll continuously on any incline, no matter how small the angle. For all other shapes, the stability is positive. On a horizontal plane or small incline it will rock about an equilibrium position. It will only start rolling continuously if the angle is large enough.
No comments:
Post a Comment