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## Homework Statement

A spherical pendulum consists of bob of mass m attached to a massless rod of fixed length R. The end of the rod opposite the bob pivots freely (in two directions) about some fixed point.

For the conical pendulum (θ=constant) case, show that the conical angle is stable for θ< θ/2. That is, show that if θ=θ

_{0}+ε then oscillates about θ

_{0}in harmonic motion. Plot the frequency of oscillation for angles 0<θ<∏/2. Comment on any interesting aspects of the curve.

g = 9.8, R = 1.8824, m = 1.96706

## Homework Equations

$$H=\frac{1}{2mR^2}\left(p_\theta^2 + \frac{1}{\sin^2\theta} p_\phi^2\right) + mgR(1-\cos\theta)$$

## The Attempt at a Solution

I don't really know how to type this out too well, but I took the derivative of H with respect to the canonical momentum in theta to get the first derivative of theta, then took the time derivative of that. this yielded

$$\ddot{\theta} = \frac{\dot{p}_\theta}{mR^2}$$

The hamiltonian to find the first derivative of p

_{θ}and plugging it in yielded

$$\ddot{\theta} = \frac{p_\phi^2}{m^2 R^4 \tan\theta\sin^2\theta}-\frac{g\sin\theta}{R}$$

and was from here unable to find an equilibrium solution

Places where pθ or p[itex]\phi[/itex] appear are meant to indicate p

_{θ}and p

_{[itex]\phi[/itex]}respectively, I just couldn't figure out how to do subscripts in latex

Moderator note: I reformatted your equations using LaTeX. Let me know if I made a mistake.

You guys are my last hope. Thanks!

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