A popular problem is the "bug sliding down a frictionless bowling ball." The problem is to find the angle, relative to the vertical, at which the bug will just leave the surface of the ball. A more modern variation of this is a skier sliding down a frictionless, large snowball. This problem involves a combination of concepts, which is why it is so popular. Those key concepts include force diagrams, circular motion, and conservation of energy. It turns out the magic angle, which works for all masses and sizes of the ball, is cos(theta) = 2/3, or theta ~ 48 degrees. Check this out to see how it works out.
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