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 = [tex] 2\sqrt{L/g}[/tex].

Homework Equations

v = [tex] \sqrt{\tau/\mu}[/tex]. (Where [tex]\tau[/tex] is the tension and [tex]\mu[/tex] 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 [tex] m(x) = \mu x [/tex]
giving [tex]\tau (x) = m(x)g = \mu g x[/tex]
so [tex] v (x) = \sqrt{\mu g x/\mu} = \sqrt{gx}[/tex]

Also [tex] L = \int^{t_0}_{0} vdt[/tex]

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!