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# Rolling on a sine curve

## Question

Suppose we have a sine (x) curve over the domain [0,pi].  Two coins will (roll/spin) starting at x=0 and go to pi.  The coins have a diameter of 1 inch.  If one starts on top of the curve and the other rolls from the bottom side of the curve, which coin would make it to pi first?

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In any case,

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The coin on the lower side of the curve will make fewer rotations and probably arrive at pi first.

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If the diameters have units, then possibly the sine wave should also have dimensions. For example, is the height of the curve 1 inch, and is the "domain" from 0 to pi inches? And if the coins are in a "race," should we make some assumptions about how they move? I mean, "the coin that moves the fastest will make it to pi first" doesn't sound like the solution to a puzzle. If the coins move so that they touch the same point on the sine wave, on opposite sides of it, they will end up in a tie. Should we assume they "spin" at constant and equal number of radians per second?

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16 hours ago, bonanova said:

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If the diameters have units, then possibly the sine wave should also have dimensions. For example, is the height of the curve 1 inch, and is the "domain" from 0 to pi inches? And if the coins are in a "race," should we make some assumptions about how they move? I mean, "the coin that moves the fastest will make it to pi first" doesn't sound like the solution to a puzzle. If the coins move so that they touch the same point on the sine wave, on opposite sides of it, they will end up in a tie. Should we assume they "spin" at constant and equal number of radians per second?

The coins spin at the same rate.  The diameter of the coins only matter in this story if we wanted to determine how much sooner one coin would finish before the other one (if there have the same diameter)

However, yes, I would think it is safe to assume that x and y are measured in inches.

If you recall from a post i made a long time ago, we saw that if two identical coins were spinning around a circle (even if the circle is defined without width) the coin on the inside of the circle would finish the circle sooner relative to the ratio of coin size to the centers of rotation that the coins spin in.

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Yes I agree. If the coin centers follow different-length paths, the shorter path will be traversed first. And since the radii are not zero, then even if the sine wave is in light-years the coin following the concave side of the curve will arrive first. (See my "In any case" post.)

In the limit as the ratio of radius to amplitude goes to zero, however, the paths of the centers (and their transit times) coalesce. I just wondered if there were a reason to give units to the diameters (and not the amplitude of the sine wave.)

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