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Total Energy of Vibration

  1. Nov 20, 2014 #1
    1. The problem statement, all variables and given/known data
    a.) Find the total energy of vibration of a string of length L, fixed at both ends, oscillating in its nth characteristic mode with an amplitude A. The tension in the string is T and its total mass is M. (HINT: consider the integrated kinetic enery at the instant when the string is straight so that it has no stored potential energy over and above what it would have when not vibrating at all.)

    b.) Calculate the total energy of vibration of the same string is it is vibrating in the following superposition of normal modes:
    y(x,t)=A1sin( xpi/L)cos(w1t) + A3sin(3xpi/L)cos(w3t- pi/4)
    (You should be able to verify that it is the sum of the energies of the two modes separately.)

    Answers:
    a.) (A^2)(n^2)(pi^2)T/4L
    b.) (A1^2 + 9A3^2)(pi^2)(T)/4L



    2. Relevant equations
    y(x,t)=A1sin( xpi/L)cos(w1t) + A3sin(3xpi/L)cos(w3t- pi/4)
    U=1/2 mu (dy/dt)^2 + T/2 (dy/dx)^2



    3. The attempt at a solution
    https://ca.answers.yahoo.com/question/index?qid=20141120113724AAqz07h [Broken]
    https://ca.answers.yahoo.com/question/index?qid=20141120113815AANH7NE [Broken]

    Thats my attempt, final answer is off ( i put the answer from textbook at the bottom of the last page), sorry couldn't figure out how to post . Any help would be greatly appreciated!

    also i just tried to get rid of the omegas using : w(n)=npi/L (T/mu)^(1/2) where mu=M/L
     
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Nov 20, 2014 #2
    Oh sorry forgot to mention I figured out A, I'm only trying to solve part b now!
     
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