Taylor

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



$$E=T+V$$

$$T=\frac{m\dot{x}^2}{2}$$

$$U_g=mgh$$

$$\Gamma=I\alpha$$



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:

$$U(\phi)=mg(l-l\cos{\phi})=mgl(1-\cos{\phi})$$



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:

$$T(\phi,\dot{\phi})=\frac{m\dot{\phi}l\sin{\phi}}{2}$$

and the potential, U, is:

$$U(\phi)=mgl(1-\cos{\phi})$$

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?