Two identical balls roll across the top of a table on parallel paths. One of the balls has to roll down into and up out of a dip in the table. The other ball rolls on the flat all the time. Which ball gets to the far side of the table first, and why?.

(In reply to Return of the Cycloid by brianjn)

Thanks for the interesting animation. If the centre of the circle moves at constant speed, the red dot clearly produces a cycloid, but I wondered whether the speed of the dot would truly match that of an object moving on an upside-down cycloid under gravity. Well, after a bit of revision, I believe it does, provided the centre of the ‘rolling’ circle moves with constant speed sqrt(gr), where r is the radius of the rolling circle.
Thus, if our object follows a complete cycle of a smooth cycloid, falling vertically from rest at a cusp and returning to the same level, then it will have travelled the same distance horizontally as a steadily moving object with speed v provided r = (v^2)/g. If we want it to get ahead then all we need is a bigger cycloid, produced by a circle with radius greater than (v^2)/g.

The trouble with this theory and others which involve ‘v’ shaped ramps etc., is that our object is moving horizontally and to suddenly change that requires a force that would be available to a bead on a wire but not to our free object. If it is to keep moving with the same horizontal speed, then we must provide it with a parabolic curve – the shape of its free trajectory. If the curvature is greater than this path the object will leave the surface. At any stage, a smooth curve in the opposite direction can return it to a horizontal path where its increased speed, because of the gain in kinetic energy, will ensure that it moves ahead of the steady object. It can then return to the original level using a mirrored parabola.

All these theories make sense for a particle on a smooth surface, but we need a rough surface to provide sufficient frictional force to change the angular momentum of a ball when it changes direction. To provide the necessary normal reaction for this, I believe the curvature will need to be slightly less than that of a parabola (more work to do here). Provided the ball rolls without slipping, no work will be done by the frictional forces so that mechanical energy will still be conserved, and the ball can be guided to a lower horizontal section which, if long enough, will give it time to catch up and overtake the steady ball.

I’ll be campaigning to lower all our motorways in this way. (That’s after the ‘banish friction’ campaign..)

Posted by Harry on 2009-11-17 16:36:52