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Easy standing wave question

  1. May 5, 2016 #1
    1. The problem statement, all variables and given/known data
    A rope, under a tension of 209 N and fixed at both ends, oscillates in a second-harmonic standing wave pattern. The displacement of the rope is given by
    y=0.49sin(πx/3)sin(12πt)
    What are (a) the length of the rope, (b) the speed of the waves on the rope, and (c)the mass of the rope? (d) If the rope oscillates in a third-harmonic standing wave pattern, what will be the period of oscillation?

    2. Relevant equations
    T=2π/w

    3. The attempt at a solution
    I have solved part a,b,c but I got part d wrong. I simply used T=2π/w
    ==> T=2π/(12π)=0.1666666...=0.167s
    And this is not the correct answer.
    Please point out where I did wrong
     
  2. jcsd
  3. May 6, 2016 #2

    haruspex

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    The given 12π is for the second harmonic. Part d asks about the third harmonic.
     
  4. May 6, 2016 #3
    I see, but is it okay to assume that the length of the rope remains the same?
     
  5. May 6, 2016 #4

    haruspex

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    Yes.
     
  6. May 6, 2016 #5
    With the same L, I got speed v=sqrt(T•L/m) where T is tension, so v remains the same too. the wavelength is changed since its now in third harmonic pattern, L=6=(3/2)λ.
    ==> λ=4
    ==> λ•f=v and from part b, v=36 is the correct answer
    ==> v=1/9=0.111...
     
  7. May 6, 2016 #6

    haruspex

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    Did you mean T=0.111?
     
  8. May 6, 2016 #7
    Yes. Sorry about the typo
     
  9. May 6, 2016 #8

    haruspex

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    Looks right.
     
  10. May 6, 2016 #9
    I checked with the answers, it is!
     
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