# 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