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Lagrangian for Applied Torque on a Hoop with Bead

  1. Oct 28, 2012 #1
    1. The problem statement, all variables and given/known data

    I included the problem as an attachment. My difficulty lies within understanding how to account for the applied torque within the Lagrangian and subsequent Euler-Lagrange equations, which is what I want to use to determine the equations of motion of the bead on the hoop through ψ''[t] and θ''[t].


    2. Relevant equations

    My physical understanding of the system tells me that the torque induces an angular velocity about the z axis which contributes to the kinetic energy of the particle. In addition, the particle's velocity along the path of the hoop contributes to the kinetic energy of the particle.

    The applied torque is:
    τ = I ψ''[t]

    The Lagrangian is naturally:
    L = T-V = total kinetic energy of the system - potential energy
    T = 1/2 m v2 = 1/2 m v12 + 1/2 m v22
    v1 = Rθ'[t]
    v2 = ψ'[t]Rcosθ
    V = -mgRcosθ

    Euler-Lagrange with q as the generalized displacement variable:
    d/dt(∂L/∂q'[t]) - ∂L/∂q = τ?

    I'm not sure if it is appropriate to set this equal to the applied torque, rather than 0 (which is what I've been used to with closed systems).

    Am I on the right track here?
     
  2. jcsd
  3. Oct 28, 2012 #2
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