Transverse Wave on a Hanging Cord

AI Thread Summary
The discussion addresses the problem of calculating the speed of transverse waves on a vertically hung cord and the time it takes for a pulse to travel along it. The speed of the wave is derived as v = √(gh), where h represents the height above the lower end of the cord. The tension in the cord varies along its length, with T = μyg, where μ is the linear density and y is the height from the bottom. The time for a pulse to travel from one end to the other is calculated as L/v. The clarification of h as a variable rather than a constant is a key point in understanding the problem.
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[SOLVED] Transverse Wave on a Hanging Cord

Problem. A uniform cord of length L and mass m is hung vertically from a support. (a) Show that the speed of tranverse waves in this cord is \sqrt{gh} where h is the height above the lower end. (b) How long does it take for a pulse to travel upward from one end to the other?

For (a), I know that the speed of a transverse wave on a cord is given by v = \sqrt{T/\mu} where T is the tension on the cord and \mu is the linear density. As far as I understand, T = mg and \mu = m/L so v = \sqrt{gL}. Now, unless h = L (which I know isn't), I don't see how h plays a role here.

The answer to (b) is just L/v obviously.
 
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Realize that the tension varies along the length of the cord.
 
Let's measure the cord from the bottom up with the bottom being y = 0 until y = L. Then tension on the cord at y is \mu y g. Is that what you mean when you wrote that the tension varies along the cord?
 
Absolutely. Using the terminology of the problem statement, T = \mu g h.
 
Ah, OK. I get it now. I thought h was some constant like L, not a variable.
 

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