Edit: Can someone change the name of the thread somehow? I accidentally posted it without changing the name.

1. The problem statement, all variables and given/known data

The question is quite long so here is a picture: http://ift.tt/1tZyKAB

2. Relevant equations


3. The attempt at a solution

If we set the zero of potential energy at origin (where the mass is at when the system is in equilibrium), then the change in gravitational potential is:

Now that we’re done with part (a), I move on to part (b), which is where I think I messed up.
The total mechanical energy of the system can be defined to be:
$$E(\phi, \dot{\phi})=T(\phi,\dot{\phi})+U(\phi)$$
Where the Kinetic energy (T) is:
and the potential, U, is:
Here is where I think I messed up mathematically; taking the time derivative of the mechanical energy, I get this:
$$\dot{E}=ml\left(\frac{\dot{\phi}\ddot{\phi}\cos{\phi}}{2}+g(\dot{\phi }\, \cos{\phi}+1)\right)$$

Did I go wrong somewhere?


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