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This is another of my favorites as well because the problem is not
really "to find the angle down the dome where the particle first
leaves the dome's surface in terms of the parameters of the problem",
rather it is "to to find the angle down the dome where the particle
first leaves the dome's surface," PERIOD.
One can show by
dimensional analysis alone that the answer can't depend on g, m, or R
as long as the object IS a particle.