Aristotle's wheel paradox • If you roll a rigid wheel a distance of its circumference, each inner circle must also go that same distance wait..but how?pic.twitter.com/M5JU06Rccv
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Resolution: • Consider the 2 points 𝐵 & 𝑆 (big/small) • Each travels 2πR horizontally (R=rad. of B) • 𝑆's Cycloid path is *shorter* than 𝐵's ⇒ The closer 𝑆 is to the center, the shorter and more direct the path ∴ 𝑆 goes 2πR horizontally b.c it takes a shorter pathpic.twitter.com/UJyfSAuKJg
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I noticed after the fact that the main gif is somewhat misleading So I plotted the cycloids along the wheel, which makes the resolution to the paradox much easier to see.pic.twitter.com/uZ4gvYEHVF
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Isn't this displacement and not distance
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Ahah!
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The apparent paradox seems to be that S, if it were rolling at the edge, wd need to complete more than 1 revolution to travel 2pi R, but here it’s only completing 1 bc it’s rotating & translating simultaneously. Your diagram shows translation relative to rotation is more than B’s
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In other words, B only rolls, while S rolls and skids some too. This is most apparent to me where they touch the bottom point of their respective circles.
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