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/snip/I sometimes wondered why physicians place so much stock in naming a complaint.
Problem:
Suppose you are allowed to change the shape of the bowl from a circle/sphere to some other concave shape that causes the resulting period of the sliding mass to actually execute true simple harmonic motion for any amplitude (within reason) when the motion is still in a vertical plane. Question: What is the shape that the bowl must have for this to work out? Extra credit: After you have found this curve/shape, is there some way, in hindsight, whereby you could have figured out the shape without doing all that work? Stipulation: The amplitude of the motion is to be measured by the path length traveled by the mass/bob for whatever path shape is to be used.
David Bowman