A Cycloid is a Tautochrone curve, meaning all points take the same time to reach the bottom regardless of initial heightpic.twitter.com/sX94kTPWll
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Is this property still true if we roll cylinders or rings instead of balls? Or is the property only for sliding point masses?
I don’t think it would hold (at least not obviously) for objects with a non-trivial moment of inertia. At least that’s my immediate response. That’s a fun/good question though
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Did you ever do the calculation? My initial guess would be that rolling doesn't matter because it would change the KE in the Lagrangian by a multiplicative constant. Maybe I'll do the calculation later when I have time.
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