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Pendulum consists of a rod of mass m attached to a light rod

  1. Aug 28, 2012 #1
    1. The problem statement, all variables and given/known data
    A pendulum consists of a uniform rod of mass m and length l hanging from the bottom end of a light rod of length l which top end is fixed to the ceiling. (see file attached)
    System moves in a vertical plane. Find equations of motion.

    Coordinates of the center of mass (X,Y)
    angles θ and ψ of the light rod and the rod of mass m with the vertical respectively.

    2. Relevant equations

    Lagrangian method

    L=T-U
    U=mgY
    T=mV2/2 + IΩ2/2

    V is the velocity of the center of mass respect to a system at rest which origin is the top end of the light rod.

    3. The attempt at a solution

    X=l/2sinψ + lsinθ
    Y=-l/2cosψ -lcosθ

    |V|2= l2/4[itex]\dot{ψ}[/itex] + l2[itex]\dot{θ}[/itex]2 + l2[itex]\dot{ψ}[/itex][itex]\dot{θ}[/itex]cos(ψ-θ)

    I relative to the top end of the rod of mass m I=ml2/3
    ω=[itex]\dot{ψ}[/itex]

    then i will plug this into L= T-U and the fin the Euler- Lagrange equations.
    but i am not sure about I and ω.

    I am confused. My first attempt was to choose same X,Y,V but I relative to the center of the rod of mass m I= ml2/2 and ω=[itex]\dot{ψ}[/itex]+[itex]\dot{θ}[/itex]
     

    Attached Files:

    Last edited: Aug 28, 2012
  2. jcsd
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