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Rotating pendulum (Lagrangian)

  1. Mar 17, 2015 #1
    1. The problem statement, all variables and given/known data rod.png
    Find the Cartesian coordinates (x, y, z) of a fixed reference frame expressed in terms of the coordinates (x', y' , z) of a rotating frame, which rotates with the horizontal rod HR. Choose the x' -axis to point along the horizontal rod in the direction OA.
    Use this to find the Cartesian coordinates (x, y, z) of the pendulum expressed in terms of θ and t

    2. Relevant equations



    3. The attempt at a solution
    I tried this:
    ##x = x' cos(\omega t)##
    ##y = y' cos(\omega t)##
    ##z = z##

    ##x' = a\hat{i}##
    ##y' = b sin(\theta)\hat{j}##
    ##z = -b cos(\theta)\hat{k}##

    So: ##\vec{r} = (acos(\omega t), bsin(\theta)cos(\omega t), -bcos(\theta)) ##

    Im later supposed to get a Lagrangian that looks like this:
    ##L =1/2mb^2\dot{\theta^2}+mab\omega cos(\theta)\dot{\theta}+1/2mb^2 \omega^2sin^2(\theta)+mgbcos(\theta)##
    but its not quite working. Any tips?
     
  2. jcsd
  3. Mar 18, 2015 #2

    Orodruin

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    Yout primed x and y coordinates should depend on both unprimed x and y coordinates in order to represent a rotation. If not, your coordinate transformation is singular at ##t = 1/\omega## (and represents a time dependent rescaling, not a rotation).
     
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