Homework Help: Finding angular frequency about the equilibrium position.

1. Apr 4, 2012

Aesteus

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. Apr 4, 2012

pymn_nzr

Last edited: Apr 4, 2012
3. Apr 4, 2012

Aesteus

But how does that help me solve for angular frequency? I have already found the equilibrium positions where U(x) is concave.

4. Apr 4, 2012

pymn_nzr

this is diagram U(x)
1-equilibrium position(STATIC)$$\left.\begin{matrix} \frac{\partial^2 U}{\partial x^2} \end{matrix}\right|_{X=x_{1},x_{2}}> 0$$

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

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

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5. Apr 4, 2012

pymn_nzr

an equilibrium is not at x=β and x=-4β

6. Apr 4, 2012

Aesteus

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.

7. Apr 4, 2012

ehild

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