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Sound & Music - Mass per unit length

  1. Feb 17, 2017 #1
    1. The problem statement, all variables and given/known data
    Suppose a harp string is tuned to middle C (C4) and is 0.56m long. If I want the tension to be 193.5 N, what mass per unit length do I need the harp string to be? Calculate your answer in kg.m

    2. Relevant equations
    Vs = sqrt(T/μ)
    V = fλ

    3. The attempt at a solution
    Having a very hard time. I looked up the frequency of a C4= 261.61 Hz. I calculated Vs as 261.61 x 1.12 = 293. Then I attempted (293 x 193.5)sq, but that number is way too high. I am stuck.
     
  2. jcsd
  3. Feb 17, 2017 #2
    Why did you multiply 293 x 193.5?
     
  4. Feb 17, 2017 #3
    I figured if Vs = sqrt (T/μ), then μ= Vs x T sq
     
  5. Feb 17, 2017 #4
    If Vs = sqrt(T/μ), then μ = T/Vs2, doesn't it?
     
  6. Feb 17, 2017 #5
    Thats what I thought initially! But when I answered 0.436 kg/m the it said it was incorrect.
     
  7. Feb 17, 2017 #6
    I don't see how you got 0.436 kg/m. I got a different answer. Can you write out what you did mathematically.
     
  8. Feb 17, 2017 #7
    Okay, I see what you did. You divided 193.5 by 293, and then you squared it. It is not (193.5/293)2. It is 193.5/(293)2.
    Or another way to do it is divide 193.5 by 293 and then divide that result by 293 again.
     
  9. Feb 17, 2017 #8
    μ = (T/Vs)sq

    T = 193.5
    Vs = 293

    193.5/291 = .66
    .664948sq = .4356 kg / m

    This answer was not correct
     
  10. Feb 17, 2017 #9
    Oh! Okay let me try that
     
  11. Feb 17, 2017 #10
    Okay finally! .0022 kg/m is correct. Can you explain to me why it is 193.3/(293) sq instead of (193.3/293)sq?
     
  12. Feb 17, 2017 #11
    Algebraically, that's how the equation turns out when you rearrange it to solve for μ.

    Vs = sqrt(T/μ)
    Square both sides of the equation to give:
    Vs2 = T/μ
    Multiply both sides of the equation by μ to give:
    μVs2 = T
    Divide both sides of the equation by Vs2 to give:
    μ = T/Vs2

    Does that make sense?
     
  13. Feb 17, 2017 #12
    That does make sense!! Thank you so much for all of your help. I really appreciate it!
     
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