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Finding angular frequency about the equilibrium position.

  1. Apr 4, 2012 #1
    1. The problem statement, all variables and given/known data

    Alright so I've got a potential energy equation U(x) = E/β^4(x^4+4βx^3-8(β^2)x^2) and U'(x) = E/β^4[4(x^3) + 12β(x^2) - 16(β^2)x] (where β and E are constants) that describes a particle of mass m which is oscillating in an energy well. I solved for where the system has equilibrium positions through simple differentiation of U(x). (an equilibrium is at x=β and x=-4β) Now I have to find the angular frequency about each equilibrium position and estimate how small the oscillations should be around the equilibrium position.

    2. Relevant equations

    N/A

    3. The attempt at a solution

    I figure that since dU/dt = ma, I can differentiate U(x), equate it to acceleration, and solve for ω. However, the equation for U(x) is rather messy, and I still don't know across which distance the particle is oscillating. Any ideas?
     
    Last edited: Apr 4, 2012
  2. jcsd
  3. Apr 4, 2012 #2
     
    Last edited: Apr 4, 2012
  4. Apr 4, 2012 #3
    But how does that help me solve for angular frequency? I have already found the equilibrium positions where U(x) is concave.
     
  5. Apr 4, 2012 #4
    this is diagram U(x)
    1-equilibrium position(STATIC)[tex]\left.\begin{matrix} \frac{\partial^2 U}{\partial x^2} \end{matrix}\right|_{X=x_{1},x_{2}}> 0[/tex]

    2-indfferent[tex]\left.\begin{matrix} \frac{\partial^2 U}{\partial x^2} \end{matrix}\right|_{X=x_{1},x_{2}}= 0[/tex]

    3-equilibrium position(not static)[tex]\left.\begin{matrix} \frac{\partial^2 U}{\partial x^2} \end{matrix}\right|_{X=x_{1},x_{2}}< 0[/tex]
     

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  6. Apr 4, 2012 #5



    an equilibrium is not at x=β and x=-4β
     
  7. Apr 4, 2012 #6
    Arghgagh ... there should be a prime in the original potential energy equation above. I corrected it.
    Thanks.

    -But the values of x = β and -4β are still correct.
     
  8. Apr 4, 2012 #7

    ehild

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    Homework Helper
    Gold Member

    The equilibrium points are correct. Write up the Taylor expansion of U(x) around both of them and stop at the second order term.

    ehild
     
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