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Homework Help: Energy of a Plucked String

  1. Apr 3, 2007 #1
    Problem. A string fixed at two ends (which are a length L apart) is pulled up at the center to a height of h. Assuming that the tension T remains constant, calculate the energy of the vibrations of the string when it is released. [Hint: What work does it take to strech the string up?]

    The work to pull the string is

    [tex]\int_0^h \frac{y}{c} \, T \, dy[/tex]


    [tex]c = \sqrt{y^2 + (L/2)^2}[/tex]

    right? And if I were to calculate the energy directly, I would need to know the frequency of vibration and the linear density of the string right?
  2. jcsd
  3. Apr 4, 2007 #2
    Nevermind. One may find the frequency given the linear density using the fact that a standing wave is produced when the string is released so:

    [tex]f = \frac{v}{\lambda} = \frac{v}{2L} = \frac{\sqrt{T/\mu}}{2L}[/tex]

    The energy is then given by:

    [tex]E = \int_0^L 2 \pi^2 f^2 D(x)^2 \mu \, dx[/tex]


    [tex]D(x) = h \sin (\pi x / L)[/tex]

    right? I should, in theory, get the same answer using this method and the method in the first post.
  4. Apr 4, 2007 #3
    I ask because the latter integral is much easier to calculate (at least for me) than the former one. For the latter one, I get [itex]\pi T h / L[/itex] as the answer.
  5. Apr 6, 2007 #4
    It just dawned on me that

    [tex]\frac{d}{dy} \sqrt{y^2 + (L/2)^2} = \frac{y}{\sqrt{y^2 + (L/2)^2}}[/tex]

    Duh! So the integral in the first post becomes [itex]T(\sqrt{h^2 + L^2/4} - L/2)[/itex]. This doesn't agree with what I posted earlier. (After a quick dimensional analysis, I realize that the energy I calculated in post #3 is wrong.) Hmm...
    Last edited: Apr 6, 2007
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