- #1
Oijl
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Homework Statement
Consider a system of one generalized coordinate theta, having the following Lagrangian equation of motion:
r and b are constants
m is mass
(1/3)mb[tex]^{2}[/tex][tex]\ddot{\theta}[/tex] = r(r+b)[tex]\theta[/tex] + r[tex]^{2}[/tex][tex]\theta[/tex][tex]^{3}[/tex] + gr[tex]\theta[/tex]
And this potential energy (if it matters):
U = mg(r+b) - mgr[tex]\theta[/tex][tex]^{2}[/tex]
There is an equilibrium point where theta is equal to zero.
Find the angular frequency of small oscillations about [tex]\theta[/tex] = 0.
Homework Equations
The Attempt at a Solution
Using the potential energy, can't I just say
U = (1/2)k[tex]\theta[/tex][tex]^{2}[/tex]
where
k = 2mgr
so that I can write
[tex]\omega[/tex] = (k/m)^(1/2)
[tex]\omega[/tex] = (2gr)^(1/2)
and call that the angular frequency?
But the problem asks me to do it the Lagrangian way.
So
[tex]\omega[/tex] = (2[tex]\pi[/tex])/[tex]\tau[/tex]
How can I find tau?