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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.