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Energy in a long inextensible string

  1. Oct 27, 2014 #1
    string.png 1. The problem statement, all variables and given/known data
    Find the maximum kinetic energy and total energy of the system if a string(with uniform linear density) of length L and mass M is oscillating as a standing wave with two fixed ends in its fundamental frequency f with amplitude 2A

    2. Relevant equations
    let x=0 and x=L be the coordinates of each end .
    Assuming the solution to the wave equation is A(cos(kx-wt)-cos(kx+wt))
    =2Asin(kx)sin(wt)
    Thus the amplitude as a function of x is 2Asin(kx)
    and kL=pi

    w=2*pi*f
    3. The attempt at a solution

    Treating the string as a series of harmonic oscillator ,
    the max.KE of each oscillator is then (1/2)[(w*2Asin(kx))^2](M/L)(dx) <--basically just half mv^2

    then , integrating the expression w.r.t. x from x=0 to x=L gives

    (M/L)(wA)^2* the integral of 2(sin(kx))^2

    =(M/L)(wA)^2* the integral of 1-cos(2kx)
    =(M/L)(wA)^2*L
    =M(wA)^2

    and by conservation of energy , max kinetic energy=the total energy of the system

    I'm not really sure if it's a correct approach
     
    Last edited: Oct 27, 2014
  2. jcsd
  3. Oct 27, 2014 #2

    collinsmark

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    Homework Helper
    Gold Member

    Yes, that looks correct to me. :)

    One thing though, the problem statement stated the frequency in terms of simple frequency, f. You should make the appropriate substitution your representation with ω = 2πf before submitting your final answer.

    But yeah, that looks to be a valid approach.
     
  4. Oct 28, 2014 #3
    alright. thank you very much
     
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