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String attached to a massless ring in both ends

  1. Jun 13, 2017 #1
    First, sorry if theres grammar mistakes,english is not my native language.
    1. The problem statement, all variables and given/known data

    Find the normal modes of a string of lenght L with a massles ring ,free to move on the y axis ,attached to each end.

    2. Relevant equations
    General wave solution: u(x,t)=A·e(kx-wt)i+B·e(-kx-wt)i
    Newton second law: F=ma
    k=w/v
    3. The attempt at a solution
    First I use Newton second law on each ring to entablish the contourn conditions, for x=0 and x=L


    Fy=m·ay=T·sen(θ)=0 , wich is 0 for being a massless ring
    (θ is the angle the string forms at x=0 with the x axis)
    for small values of θ we can take tanθ instead of sinθ, and since the definition of derivate is dy/dx=tanθ replacing above for y =u(x,t) and evaluated in x=0 we obtain
    ∂u/∂x (on x=0)=0 and the same for x=l

    Now If we suppose a wave solution
    u(x,t)=A·e(kx-wt)i+B·e(-kx-wt)i
    and aply the CC I obtain
    kn=n·π/l and u(x,t)=A·e-iwt·2cos(n·π·x/L)

    My doubt is if this is a correct answer, since I obtain the same k for a string with both ends fixed. Also for n=1
    u(0,t)=A and u(L,t)=-A wich doesnt sound right to me for λ1=2·L/π
     
  2. jcsd
  3. Jun 13, 2017 #2

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Hello Javier, :welcome:
    Why not ? it satisfies the boundary conditions

    Yes, only there you have a sine instead of a cosine. Google 'normal modes open pipe' and check a few pictures.
     
  4. Jun 13, 2017 #3
    Got it now, thanks a lot
     
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