Homework Help: Wave velocity in a free-hanging rope

1. Sep 23, 2010

CharliH

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

A uniform rope of length L hangs freely from the ceiling. Show that the time for a transverse wave to travel the length of the rope is t0 = $$2\sqrt{L/g}$$.

2. Relevant equations

v = $$\sqrt{\tau/\mu}$$. (Where $$\tau$$ is the tension and $$\mu$$ the linear density of the rope.)

3. The attempt at a solution

Set up axes so that the rope is parallel to the x-axis, with the bottom of the rope at the origin.

Let m(x) represent the mass of the rope below x. Then $$m(x) = \mu x$$
giving $$\tau (x) = m(x)g = \mu g x$$
so $$v (x) = \sqrt{\mu g x/\mu} = \sqrt{gx}$$

Also $$L = \int^{t_0}_{0} vdt$$

I can see that velocity is a function of time and that integrating will give me something at least similar to the required equation, but I can't figure out how to get v in terms of t. Or maybe I should be getting x in terms of t. I couldn't find that either, though.

2. Sep 23, 2010

rl.bhat

v = dx/dt = sqrt(gx)

dx/sqrt(gx) = dt.

Now find the integration and put the limits x = 0 to x = L

3. Sep 23, 2010

CharliH

Ohhh, I get it now. Thanks!