Mode
Descent times
Line
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Arc
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Parabola
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Cycloid
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Analytic cycloid time / winner
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Beads integrate s'' = g dy/ds along each curve (frictionless). Cycloid x=a(t-sin t), y=a(1-cos t); T = t_B sqrt(a/g). Tautochrone: released from different heights on a cycloid, all reach the bottom together (T = pi sqrt(a/g)).