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Wave on string, bdry cond is mass on spring

  1. Sep 23, 2012 #1
    1. The problem statement, all variables and given/known data
    A semi-infinite string (constant tension T and density ρ) lies along the positive
    x-axis. The motion of the string is described by the linear wave equation
    \begin{equation}
    h_{xx}-c^2h_{tt}=0
    \end{equation}
    with
    c² ≡ T/ρ, where h(x,t) is the transverse displacement. Attached to the end of the string at
    x = 0 is a mass m. Attached to the mass in the vertical direction is a spring with stiffness
    constant k that has its other end attached to a rigid surface. The vertical displacement of
    the mass m is designated by ξ (t) ≡ h(0,t). When the system is in equilibrium (i.e.,
    when the spring is unextended and everything is at rest) ξ (t) ≡ 0. The radian frequency
    of the free oscillator (when the string is not attached) is σ = k /m .
    (a) By applying Newton's law to the motion of the mass at x = 0, find the boundary
    condition on the string at x = 0. That is, find the equation of motion of the mass in
    terms of ξ (t) . Ignore horizontal forces on the mass, and ignore gravity.
    [HINT: the boundary condition should not have any terms showing h(0,t).]
    (b) Prove that the total energy of the mechanical system (oscillator and string) is constant.
    (c) Suppose that at t = 0 the initial conditions are


    3. The attempt at a solution

    a)
    There are two forces (neglecting $\vec{g}$) acting on the mass: one due to the displacement (spring) and one stemming from the tension in the string.
    \begin{align}
    F_{net}
    &=
    \sum_i F_i\\
    f_{net}
    &=
    \sum_i f_i\\
    \ddot{\xi}
    &=
    -\sigma^2 \xi
    + c^2 h(0,t)_{xx}
    \end{align}

    b)
    ?
    c)
    ?

    ....aaand that is about as much as i can come up with..

    i played around with seperating variables a bit.. i don't even know what type of solution to expect.. something like
    \begin{equation}
    h(x,t)=e^{-At} (B \sin(kx-\omega t)+ C \cos(kx-\omega t))
    \end{equation}
    ?
    obviously
    \begin{equation}
    \sigma \neq \omega
    \end{equation}

    please help!!!!!!!!!! i've been staring at the problem for 4 hours
     
  2. jcsd
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