A rather interesting type of coupled oscillator.

[dHannibal]

Homework Statement

The problem can be found here. http://wopho.org/dl.php?id=17&dirfile=selection-problem/helical_rope.pdf" I am attempting to solve part 3.

Homework Equations

The Lagrangian of the system is: $$L= \frac{m\dot{x}^2}{2}+\frac{mr^2\dot{\theta}^2}{2}-k \left( x^2+(r\theta)^2-2x_0\sqrt{x^2+(r\theta)^2} \right)$$
Also,the equations of motion are:
$$\ddot{x}=\frac{2k}{m} \left(\frac{x_0}{\sqrt{x^2+(r\theta)^2}} -1 \right)x$$
$$\ddot{\theta}=\frac{2k}{m} \left(\frac{x_0}{\sqrt{x^2+(r\theta)^2}} -1 \right) \theta$$

The Attempt at a Solution

I need to solve the equations of motion for $$x(t)$$ and $$\theta (t)$$. First I tried assuming $$r \theta << x$$ but it leads to equations of motion of the form

$$\ddot{x}=\frac{2k}{m} \left(\frac{x_0}{x+\frac{(r \theta)^2}{2x}} -1 \right)x$$
which is not particulaly useful. Pointing out that $$\frac{\ddot{x}}{x} = \frac{\ddot{\theta}}{\theta}$$ I tried assuming solutions of the form $$x= Ae^{(iwt + \phi)}$$ and $$\theta= Be^{(iwt + \phi)}$$ but I was again unsuccessful. I was stuck at this point.
Thank you.

 

[tiny-tim]

hi dHannibal!

try putting x'' = v dv/dx and integrating

 

[dHannibal]

Hi tinytim,
but how do i get rid of $$\theta$$'s in the square root when integrating?