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BruceW

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[tex]V = \frac{1}{2}k(r-l)^2 - mgrcos(\theta)[/tex]

Now I'm going to talk about the 2d case, because the equations are easier. So the kinetic energy of the object is given by:

[tex] KE = \frac{1}{2}m(\dot{r}^2 + r \dot{\theta}^2 )[/tex]

And now, we can use the Euler-Lagrange equations to find out the laws of the system:

[tex]-grsin(\theta) = \frac{d(r \dot{\theta})}{dt} [/tex]

[tex]mr \dot{\theta} - k(r-l) + mgcos(\theta) = m \ddot{r} [/tex]

And there is also the equation for the conservation of energy, which simply says that the kinetic energy plus the potential energy is conserved.

So, the equations are a bit complicated. We could also make the small angle approximation, which would make [itex]sin(\theta) \rightarrow \theta[/itex] and [itex]cos(\theta) \rightarrow 1 - \frac{1}{2} \theta^2[/itex] But it would still look quite complicated.

You could use these equations for a simulation on computer, and that would show the kind of trajectory to expect. And maybe there is a way to do stability analysis, which would show that certain trajectories are more stable than others, I'm not sure..

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