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We know that a point on the rim of a rolling body traces out a cycloid during one turn. Why is the length of the cycloid a constant = 8r. Shouldn't length of cycloid depend on speed of rolling body. For example, a faster body should trace out a cycloid which has a longer length than a slower body. What is the flaw in my reasoning?

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Why do you feel the arc length should depend on speed? You ask what the flaw is in your reasoning, but you don't supply any reasoning. – BMS Dec 1 '13 at 15:46
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If you wanted to measure the arc length swept out by a point on the rim of a wheel for a set amount of time, you would be correct; the faster one would cover more length in the same amount of time. It's unclear to me, but this may be the reason you believe the distances should be different.

However, the "arc length of a cycloid" $S=8r$ is calculated for the specific case of one period, aka one revolution of the wheel. The faster wheel will complete one revolution in less time than the slower one. The time intervals are different. Thus, the arc lengths don't have to be different.

To see that speed really doesn't matter, one can realize that this is a purely geometric problem. It's analogous to saying that an object moving in a circle will travel the same distance $2\pi r$ in one period, regardless of its speed. A faster one will travel in less time, but traverse the same distance in one period.

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Thus, the arc lengths don't have to be the same - shouldn't that read Thus, the arc lengths don't have to be different? – Christoph Dec 1 '13 at 16:20
@Christoph Yes, thank you. – BMS Dec 1 '13 at 16:21

Suppose you took a video of a the rolling body and then played it in slow motion. This is analogous to having the rolling body move slower. Do you really expect a different shape to emerge if you slow down the video?

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