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Coupled Pendulum/Mass on spring

  1. Mar 8, 2009 #1
    1. The problem statement, all variables and given/known data

    A mass M1 sits on a level frictionless surface attached to a spring of stiffness k. Mass M2 is supported from M1 by a string of length [tex]l[/tex]. The horiztonal displacement from the equilibrium position of M1 is x1; the horizontal displacement of the pendulum from the wall where the spring is attached is x2; and the vertical displacement of the pendulum is y2. Find the exact equations of motion for x1, x2, and y2. Then, show that the equations can be reduced, using the small angle approximation (sin(x) ~ tan(x) ~ x) and the fact that
    [tex]\ddot{y}_{2}<<g[/tex]

    to:

    [tex]M_{1} \ddot{x}_{1} = -k x_{1} + M_{2} \frac{g}{l} \left(x_{2} - x_{1}\right)[/tex]
    [tex]M_{2} \ddot{x}_{2} = - M_{2} \frac{g}{l} \left(x_{2} - x_{1}\right)[/tex]


    2. Relevant equations



    3. The attempt at a solution

    So far, I've attempted to derive the exact equations for motion:

    [tex]M_{1} \ddot{x}_{1} = -k x_{1} + M_{2} g \left(sin\theta cos\theta \right) [/tex]
    [tex]M_{2} \ddot{x}_{2} = -M_{2} g \left(sin\theta cos\theta \right) [/tex]
    [tex]M_{2} \ddot{y}_{2} = -M_{2} g \left(sin^{2} \theta \right) [/tex]

    where theta is the angular displacement of the pendulum from the equilibrium position. In the first equation, the force terms on the right come from the spring and the horizontal component of the tension in the string connecting the two masses (which is the component of the weight of the pendulum pointed parallel to the string). In the second and third equations, the force terms are just the horizontal and vertical components of the component of the pendulum's weight pointed in the direction of its motion. I find it would be much easier to just derive the equations of motion in terms of the angle theta, and then use the small angle approximation, but I have to do it from the equations I just gave. I'm just having trouble making the leap to the simplified equations. Any ideas?
     
  2. jcsd
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