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Re: [Phys-L] quickest route



I wrote:

As David Bowman points
out, analyzing a realistic rolling motion would be messier.

On 07/30/2013 02:40 PM, Bill Nettles wrote:
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 that for the globally-optimal route, the
initial motion is purely vertical, so for any finite
coefficient of friction the ball will /slip/. The
subsequent transition from slipping to non-slipping
will be what I call messy.

Similarly, the piecewise-linear ("ramps") problem that
I posed involved a vertical segment, so the same mess
appears again. I stand by what I wrote. If the ball
is allowed to roll, there is no simple proportionality
between center-of-mass KE and rotational KE.

One simple way to minimize the mess is to formulate
it as a "skateboard" problem, where a large mass rides
on tiny wheels. In other words, the effective radius
of gyration is very small. Another approach is to
imagine the simple frictionless sliding motion of a
block or of a bead on a wire.

==========================

In the following we do /not/ restrict attention to linear
ramps.

Consider the globally optimal route from A to B. Obviously
B must be lower than A, and indeed the entire route
must be lower than A. The interesting thing is that
for /some/ conditions, part of the route is even
lower than B, which means the vertical component of
the motion is non-monotonic. (Note that this feature
has been missing from the piecewise-linear "ramp"
models mentioned in this thread.)

The question is: What are the necessary and sufficient
conditions for which the optimal route spends part of its
time below B?

The usual spherical-cow approximations apply: Uniform
gravitational field, negligible friction, et cetera.