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Re: [Phys-L] quickest route



Regarding Bill N's comment:

Not really that messy if one rolls on plane surfaces. Because
the actual value of g falls out of the problem, any object
which accelerates down the plane like K*g* sin(theta), where K
is some constant depending on the details, and K*g for the
straight down part will have equal times at tan(theta) = 0.75
or theta = 36.87... Spheres or disks which roll without
slipping have such an acceleration.

The problem is more complicated for any path that includes a very
steep section and a very shallow section. The reason is that any
realistic round object will roll without slipping on sufficiently
shallow section (at least initially if it wasn't already slipping
before) and will definitely slip on a sufficiently steep section.
This means that for any such path there will probably be a
transition between skidding and rolling without skidding. The
scalar acceleration is different in the two cases, and this affects
the timings. Also, there is no way to know just where the
transition will occur without knowing more about the details of the
interfacial contact and the internal mass distribution of the round
object. The transition from skidding to rolling without skidding
is more complicated than the transition from rolling without
slipping to skidding. JD's counterexample of an L-shaped path has
both extremal situations making up 100% of the same path.

Also, for any curved path the normal force exerted on the object by
the track depends on the local speed of the object and the local
radius of curvature (because of a centripetal/centrifugal term) and
this affects the crossover angle from rolling to skidding, as well
as affecting the strength of the kinetic friction when skidding
occurs, and this affects the local scalar acceleration when there
is skidding. Not only that, but for the skidding case the
functional form of the scalar acceleration is not proportional to
a pure sine function, even for an uncurved section, because of the
presence of the kinetic frictional force (which for a typical
simple model is proportional to the normal force, and thus has a
cosine dependence on track angle).

Even if for some path (such as for a straight line path) there is
no transition there may be one for a comparison path, and this
affects the timing of the other path and may affect the answer as
to which one is traversed faster.

The only way to avoid these complications is to specifically state
up front in the problem statement either that the object always
slides frictionlessly on all paths, or that it always rolls
without slipping, over all parts of all paths (thus forbidding
any transition between the different kinds of motions and the
attendant complications). If the latter situation is chosen then
even vertical stretches of some paths must have the ball/object
rolling without slipping. Not only JD's counterexample, but even
the (supposedly) extremal brachistochrone cycloid itself has such
vertical sections. I suppose this rolling-on-a-vertical-(or-
very-steep)-surface situation could be realized with a
ferromagnetic ball and/or track, but that would probably invite
new complications caused by magnetic field inhomogeneities and
dissipation from the eddy currents generated as the ball gets
rolling with sufficient speed.

There are also other potential complications, like the ball flying
off of the track for situations where the outward normal force on
the ball attempts to become negative. But this later complication
would only be a concern for the convex-shaped track (which would
probably be already considered a loser relative to the other two
tracks). And if the ball flies off of a convex track it would
likely land beyond the track's terminus and never reach the proper
endpoint, thus more strongly guaranteeing its third place ranking.

David Bowman