Re: When Physical Intuition Fails

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding John Clement's example problems with counterintuitive
solutions:

> The recent issue of AJP posed a problem which NO physics professors
> (20) or students (67) could solve.

This is hard to believe (regarding the professors, not necessarily,
the students).

> I wonder if anyone on this list can solve it?
> Please no peeking at the answer key.  I have already peeked so I am
> disqualified.
>
> Ignore the retarding effect of air resistance.  A rigid wheel is
> spinning with an angular speed of W0 about a frictionless axis.  The
> wheel drops on a horizontal floor, slips for some time, and then
> rolls without slipping.  After the wheel starts rolling without
> slipping, the center of the mass speed is Vf.  How does Vf depend on
> the kinetic coefficient of friction mu between the floor and the
> wheel?

Like John Mallinckrodt (and I suspect many others) I've used various
variations on this problem as in-class examples and as test questions
for many years.  The versions I like to use refer to bowling balls
launched down a very long alley with both top spin and, especially,
back spin, and to derive formulae for the final velocity, down range
distance where the sliding stops, criteria for the ball reversing its
direction of motion in the back spin case, for the ball to just stop
when the sliding stops, for the ball to return to the initial
starting place and then stop sliding there, etc., etc.

> Well the third article "When physical intuition fails" in the Nov.
> issue is the one posing the problem.  The question was posed in a
> quiz to the students and was given in a 15 min oral interview to the
> faculty.  No hints were given.  The article is very interesting and
> shows how seemingly simple problems can even stump the experts.  Of
> course the author is trying to make the case that explicitly
> teaching problem solving is important.  I doubt that it will
> convince the skeptics.  I was already convinced.  The article may be
> justly called a stealth PER article.

Who are these "skeptics" that need to be convinced that even experts
can be stumped on occasion by simple problems, or that problem
solving skills are important in physics?

> Incidentally the UMass Amherst PER group had a similar problem that
> stumped many experts and students alike.  It was a car rolling down
> a slight incline vs a car rolling into a hole and then to the same
> final location.  It was surprising how many were surprised to see
> that the longer path yielded a shorter time.  They showed the result
> with movies and simulations.
>
> We will see what the reaction is.  Maybe everyone who can solve it
> has already seen the problem before?

There is a very good chance that very many on this list know about
the brachistochrone problem since it has been around for literally
hundreds of years.  Galileo showed by explicit demonstration
(remember, he loved to roll things down ramps) that the cycloid-
shaped path is the minimal time path.  The problem was solved
analytically by J. Bernoulli over 300 years ago and is a classic
(probably *the* classic) example of the application of the
calculus of variations.

A point to keep in mind about this problem is that the over all shape
of the minimal time path depends on the ratio of the vertical height
of descent between the endpoints to the horizontal range between the
endpoints because this ratio determines the total parameteric
rotation angle of the cycloid.  For instance, if the net height of
descent is greater than 2/[pi] times the horizonal range then the
minimal time path does *not* dip below the level of the final
endpoint and there is no dip or "hole".

A favorite problem of mine that seems to have a counterintuitive
solution (in that, Like John's/AJP's wheel example, the solution
doesn't depend on some of the parameters that one would naively think
it would using an uninformed physical intuition) is the case of a
particle sliding off of the top of a frictionless hemispherical dome
in the presence of a gravitational field and assuming the particle's
slide was initiated from nearly rest from the top of the dome where
only enough of an initial nudge was given to break the initial
balancing symmetry that it would have if it really was at true rest
at the true top.  Since the initial equilibrium point is unstable
it will, in practice, not be a problem to break the symmetry.  (In
fact the uncertainty principle puts a bound on the balancing time
anyway.)  The problem is to find the angle down on the dome where
the particle first leaves the dome's surface in terms of the
parameters of the problem.

David Bowman

This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.