Quick Lagrangian of a pendulum question

  1. 1. The problem statement, all variables and given/known data

    Use the E-L equation to calculate the period of oscillation of a simple pendulum
    of length l and bob mass m in the small angle approximation.

    Assume now that the pendulum support is accelerated in the vertical direction at a rate
    a, find the period of oscillation. For what value of a the pendulum does not
    oscillate? Comment on this result.

    2. Relevant equations



    3. The attempt at a solution

    I've got the first bit:
    L=(m/2)(l^2)(dθ/dt)^2-mgl(1-cosθ)
    E.O.M.: d2θ/dt2+(g/l)sinθ=0
    d2θ/dt2+(g/l)θ=0 in the small angle approximation,
    which is S.H.M. with ω^2=√(g/l) (though I'm not sure about this as there's no minus sign in the E.O.M.) so T=2pi√(l/g)

    For the next bit, I just need help setting up the equations:
    So the generalized coordinates are θ and a.
    Are the following correct?:
    x=lsinθ
    y=-lcosθ+at
    (taking the origin as the point from which the pendulum is swinging)
     
    Last edited: Mar 5, 2012
  2. jcsd
  3. jambaugh

    jambaugh 1,808
    Science Advisor
    Gold Member

    Careful here, [itex]\omega[/itex] is that angular velocity in radians per time unit. Your T is the time to travel one radian (of the oscillatory cycle) not time per cycle. You need to multiply by [itex]2 \pi[/itex].
     
  4. Thanks for pointing that out, I'll correct that in the first post. :-)
     
  5. Any help would be great. :D
     
  6. Should y be -lcosθ+0.5at^2 instead?
     
  7. I like Serena

    I like Serena 6,194
    Homework Helper

    Yes.

    No.
    "a" is not a coordinate.
    Actually only θ is a generalized coordinate since the y coordinate of the support is constrained to be y=0.5at^2.
    The angle θ is the only freedom that the system has.
     
  8. Great, I've got it now.

    Thanks! :-)
     
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