Wave velocity in a free-hanging rope

[CharliH]

Homework Statement

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 = $$2\sqrt{L/g}$$.

Homework Equations

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

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.

 

[rl.bhat]

v = dx/dt = sqrt(gx)

dx/sqrt(gx) = dt.

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

 

[CharliH]

Ohhh, I get it now. Thanks!