Analytical mechanics question.

  • Thread starter peripatein
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  • #1
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Hi,

Homework Statement


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


Homework Equations





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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Answers and Replies

  • #2
haruspex
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Do I now substitute cosθ≈1-1/2[itex]\dot{θ}[/itex]2 and sinθ≈θ
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.
 
  • #3
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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)?
 
  • #4
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Wait, dimensional analysis indicates I am wrong, doesn't it?
 
  • #5
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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)?
 
  • #6
haruspex
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I'd appreciate your feedback on the following attempt:
V = -mgR(cosθ + √3sinθ) ≈ -mgR(1 - 0.5θ2 + √3θ)
First, is that the correct approach?
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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