1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Analytical mechanics question.

  1. Nov 19, 2013 #1
    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?

    Attached Files:

    • 2.JPG
      File size:
      3.6 KB
  2. jcsd
  3. Nov 19, 2013 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Discussions: Analytical mechanics question.
  1. Analytical question (Replies: 0)

  2. Mechanics Question (Replies: 6)