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Marion and Thornton Dynamics Problem 7-20

  1. Jun 5, 2017 #1
    1. The problem statement, all variables and given/known data
    circular hoop is suspended in a horizontal plane by three strings, each of length l, which are attached symmetrically to the hoop and are connected to fixed points lying in a plane above the hoop. At equilibrium, each string is vertical. Show that the frequency of small rotational oscillations about the vertical through the center of the hoop is the same as that for a simple pendulum of length l.



    2. Relevant equations
    L = T - U

    3. The attempt at a solution
    NOT HOMEWORK SELF LEARNING

    OK my main problem here is understanding how this is happening if the support is fixed in a plane and length stays l and hoop rotates horizontally then the height of the center of mass should not change
    I need help clarifying the picture here what am I missing
    From there it's straightforward to set up the Lagrangian
     
  2. jcsd
  3. Jun 5, 2017 #2
  4. Jun 5, 2017 #3
    OK thanks again
    but I didn't sense from this the the center of mass of the hoop has risen
    so U=O?
     
  5. Jun 5, 2017 #4
    you are not obliged to understand anything a priori. Just write equations of constraints
     
  6. Jun 5, 2017 #5
    My calculation gives the follows. If##\psi## is a small angle of hoop's rotation then the height of the center of mass is ##\frac{r^2}{2l}\psi^2+o(\psi^2)##, here ##r## is hoop's radius
     
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