1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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]

    where

    [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]

    where

    [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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Energy of a Plucked String
Loading...