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

A uniform rope of lengthLhangs freely from the ceiling. Show that the time for a transverse wave to travel the length of the rope is t_{0}= [tex] 2\sqrt{L/g}[/tex].

2. Relevant equations

v = [tex] \sqrt{\tau/\mu}[/tex]. (Where [tex]\tau[/tex] is the tension and [tex]\mu[/tex] 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 [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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Wave velocity in a free-hanging rope

**Physics Forums | Science Articles, Homework Help, Discussion**