- #1
dHannibal
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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: [tex]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) [/tex]
Also,the equations of motion are:
[tex]\ddot{x}=\frac{2k}{m} \left(\frac{x_0}{\sqrt{x^2+(r\theta)^2}} -1 \right)x [/tex]
[tex]\ddot{\theta}=\frac{2k}{m} \left(\frac{x_0}{\sqrt{x^2+(r\theta)^2}} -1 \right) \theta [/tex]
The Attempt at a Solution
I need to solve the equations of motion for [tex]x(t)[/tex] and [tex] \theta (t) [/tex]. First I tried assuming [tex] r \theta << x [/tex] but it leads to equations of motion of the form
[tex]
\ddot{x}=\frac{2k}{m} \left(\frac{x_0}{x+\frac{(r \theta)^2}{2x}} -1 \right)x
[/tex]
which is not particulaly useful. Pointing out that [tex] \frac{\ddot{x}}{x} = \frac{\ddot{\theta}}{\theta} [/tex] I tried assuming solutions of the form [tex] x= Ae^{(iwt + \phi)} [/tex] and [tex] \theta= Be^{(iwt + \phi)} [/tex] but I was again unsuccessful. I was stuck at this point.
Thank you.
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