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A straight line may be the shortest 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 between two points, but a "Brachistochrone" curve is the path of least timepic.twitter.com/aWPoz9X6i9
A "Cycloid" curve is one for which all objects take the 𝑠𝑎𝑚𝑒 amount of time to reach the bottom • If this seems counterintuitive, remember that 𝑚𝑔ℎ=½𝑚𝑣², and so objects starting higher will have a larger final velocitypic.twitter.com/EV4ZzT6YFo
A Cycloid curve is also called a "Tautochrone" (meaning equal time) Mathematically, the Brachistochrone and the Cycloid are the same curve, but they arise from slightly different but related problems (equal time & least time) Brachistochrone ⇔ Cycloid Tautochrone ⇔ Cycloid
We learn in high school that the "period of a pendulum is independent of its amplitude"--i.e that it's isochronus. • This isn't quite true; it's only true for *small* angles θ • In reality, a pendulum's period is only independent of its amplitude if it moves along a cycloidpic.twitter.com/d1Vh1VlJ9u
gif credits: Brachistochrone: Robert Ferréol (Wikipedia) Tautochrone: Claudio Rocchini (Wikipedia) Pendulum : Rem088roy (Wikipedia)
A Cycloid is the curve traced out by a point on a rolling wheelpic.twitter.com/3BgmeSPDQF
gif credit: Cycloid rolling (last gif): IchibanPL (Wikipedia)
that is an interesting question
Only applies if gravity (and acceleration) in play, right? Rolling balls along grooves on a table, for example, now the line is fastest again
yes.. but there’s nothing special about g=9.8m/s^2.. it’s the fact that it’s (1) downward and (2) constant
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