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Analytical mechanics question.

  1. Nov 19, 2013 #1
    Hi,
    1. The problem statement, all variables and given/known data
    I was given the setup in the attachment and was asked to find the angular frequency of small oscillations around the equilibrium. m1=m; m2=√3m


    2. Relevant equations



    3. The attempt at a solution
    I have found L = 1/2*(3+√3)*mR2[itex]\dot{θ}[/itex]2 + mgRcosθ+√3mgRsinθ
    and the point of equilibrium to be at tgθ=m2/m2=√3
    Do I now substitute cosθ≈1-1/2[itex]\dot{θ}[/itex]2 and sinθ≈θ
    and then write down Euler-Lagrange?
     

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  3. Nov 19, 2013 #2

    haruspex

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    The approximation you're thinking of is cosθ≈1-1/2[itex]\θ[/itex]2 for small θ. It's not with a [itex]\dot{θ}[/itex] in it, and it's not for what may be not a very small θ.
    If θ is defined by tanθ=m2/m2=√3, you want to consider a small perturbation dθ from there. Try putting θ+dθ in your torque equation.
     
  4. Nov 19, 2013 #3
    I am not using any torque equations. I found the Lagrangian and was now thinking of using the Euler-Lagrange relation. In any case, could it be that k=second partial derivative of potential at point of equilibrium=2mgR
    and hence angular frequency is sqrt(k/m)=sqrt(2gR)?
     
  5. Nov 19, 2013 #4
    Wait, dimensional analysis indicates I am wrong, doesn't it?
     
  6. Nov 21, 2013 #5
    I'd appreciate your feedback on the following attempt:
    V = -mgR(cosθ + √3sinθ) ≈ -mgR(1 - 0.5θ2 + √3θ)
    First, is that the correct approach?
    Second, do I now subsitute my θ of equilibrium in ∂2V/∂q2 to get k in ω2=k/m?
    Third, how do I find m in ω2=k/m? Is it by substituting my θ of equilibrium in the approximation -mgR(1 - 0.5θ2 + √3θ)?
     
  7. Nov 21, 2013 #6

    haruspex

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    No, you didn't understand what I wrote before.
    θ cannot be assumed to be small, so you cannot use those approximations. Find the equilibrium value of θ, then express θ as that value plus a small perturbation angle. Then you can use approximations for trig functions of the small perturbation.
     
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