# Homework Help: Simple cycloid time calculation problem

1. May 20, 2007

### maverick280857

1. The problem statement, all variables and given/known data

Consider a wire bent into the shape of the cycloid

$$x = a(\theta - \sin\theta)$$
$$y = a(\cos\theta -1)$$

If a bead is released at the origin and slides down the wire without friction, show that $\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 $$\frac{1}{2}mv^{2} = mg(2a)$$ or $$v^{2} = 4ga$$ For the point at the bottom, [itex]\theta = \pi$. So, the arc length is

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

$$v = \frac{ds}{dt}$$

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

2. May 20, 2007

### maverick280857

Okay I got it. :tongue2: