Marion and Thornton Dynamics Problem 7-20

[MARX]

Homework Statement

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.

Homework Equations

L = T - U

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
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[zwierz]

 

[MARX]

OK thanks again
but I didn't sense from this the the center of mass of the hoop has risen
so U=O?

 

[zwierz]

you are not obliged to understand anything a priori. Just write equations of constraints

 

[zwierz]

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