(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider a wire bent into the shape of the cycloid

[tex]x = a(\theta - \sin\theta)[/tex]

[tex]y = a(\cos\theta -1)[/tex]

If a bead is released at the origin and slides down the wire without friction, show that [itex]\pi\sqrt{a/g}[/tex] is the time it takes to reach the point [itex](\pi a, -2a)[/tex] at the bottom.

2. Relevant equations

(See below)

3. The attempt at a solution

Energy conservation gives

[tex]\frac{1}{2}mv^{2} = mg(2a)[/tex]

or

[tex]v^{2} = 4ga[/tex]

For the point at the bottom, [itex]\theta = \pi[/itex]. So, the arc length is

[tex]s = \int_{0}^{\theta}\sqrt{\left(\frac{dx}{d\theta}\right)^{2} + \left(\frac{dy}{d\theta}\right)^{2}}d\theta[/tex]

[tex]v = \frac{ds}{dt}[/tex]

How do I get rid of the [tex]d\theta/dt[/tex]? I know I'm missing something here...

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# Homework Help: Simple cycloid time calculation problem

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