Finding angular frequency about the equilibrium position.

[Aesteus]

Homework Statement

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.

Homework Equations

N/A

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?

 

[pymn_nzr]

equilibrium position$$\rightarrow \frac{\partial U}{\partial x}=0$$



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

 

[Aesteus]

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

 

[pymn_nzr]

Well I'm trying to find the angular frequencies of the particle at its equilibrium positions.
I'm not quite sure where you go that quadratic equation. Could you explain in more detail?

Thanks

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

 

[pymn_nzr]

U(x) = E/β^4[4(x^3) + 12β(x^2) - 16(β^2)x] (an equilibrium is at x=β and x=-4β)

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

 

[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.

 

[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