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Transverse Wave on a Hanging Cord

  • Thread starter e(ho0n3
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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 [itex]\sqrt{gh}[/itex] 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 [itex]v = \sqrt{T/\mu}[/itex] where T is the tension on the cord and [itex]\mu[/itex] is the linear density. As far as I understand, T = mg and [itex]\mu = m/L[/itex] so [itex]v = \sqrt{gL}[/itex]. 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.
 

Answers and Replies

  • #2
Doc Al
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Realize that the tension varies along the length of the cord.
 
  • #3
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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 [itex]\mu y g[/itex]. Is that what you mean when you wrote that the tension varies along the cord?
 
  • #4
Doc Al
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Absolutely. Using the terminology of the problem statement, [itex]T = \mu g h[/itex].
 
  • #5
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Ah, OK. I get it now. I thought h was some constant like L, not a variable.
 

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