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Homework Help: Mass spring system

  1. Dec 3, 2014 #1
    1. The problem statement, all variables and given/known data
    Starting from system of springs and masses (on picture), fina a force in x direction which n-th mass acts on n+1st mass, if harmonic wave ψ(x,t) is traveling in system.
    In equilibrium every mass is compressed i acts with force of F0 on masses.
    Consider a case in boundary of continuum (a→0)
    a is distance from first to next mass.

    2. Relevant equations
    Second Newton's law, F=m*a
    ψ(x,t)= Asin(ωt -kx)
    3. The attempt at a solution
    Equation of motion for n-th mass:
    md2xn/dt2= k[(xn+1-xn) -n*a] -k(xn - xn-1)-n*a]

    Analogous , for n+1 mass we have
    md2xn+1/dt2= k[(xn+2-xn+1) -n*a] -k(xn+1 - xn)-n*a]
    Usually, we guess the soulution. For standard harmonic oscillator, it was x(t) = A cos (ωt + φ).
    But now, there is also a wave in here. What to do with it?
    What is his part in this problem?
    Do i try to guess soultion also?
    Should i try to find a force from as from above equations or somehow different?
    What is phyisical meaning of " springs are in equilibrium and every spring is compressed and acts on mass with force of F0?
    Does that withdraws an driven oscillator?
  2. jcsd
  3. Dec 3, 2014 #2


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    I suggest starting by replacing all of the xs by the sum of the equilibrium positions and the deviation from the equilibrium positions. This is what is going to be represented by your harmonic wave ψ.
  4. Dec 3, 2014 #3


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    The information about F0 establishes a pre-load in the system. The equilibrium position has an amount of compression in the string equal to F0.

    I suggest that you draw several FBDs and write the equations of motion for each mass, one by one, remembering that the springs are pre-loaded.
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