Rod-Spring System Homework: Verifying Parts a) & b)

[ghostfolk]

Homework Statement

I'm really looking for a verification on parts a) and b), but I'll add what I did with part c) without going to into too much detail. I'm posting this question mainly due to part d). I feel that I have every part before this right, but I'm not getting any symmetric matrices.

Homework Equations

$T=\frac{1}{2}mv^2$

The Attempt at a Solution

a)$U=\dfrac{mgL}{2}\cos{\phi}+\dfrac{k}{2}(x^2+L^2(\cos{\phi})^2)$
b)$x_{cm}=x-\dfrac{L}{2}\sin(\phi)$, $\dot{x_cm}=1-\dfrac{L}{2}\dot{\phi}\cos{\phi}$
$z_{cm}=\dfrac{L}{2}\cos(\phi)$, $\dot{z_cm}=-\dfrac{\dot{\phi}}{2}L\sin(\phi)$
$T=\dfrac{mL^2\dot{\phi}^2}{6}+\dfrac{m}{2}(1-L\dot{\phi}\cos{\phi})$
c) I know for part c) we can just use the Lagrangian of the system, then find the Euler-Lagrange and let the acceleration in the $\phi$ and $x$ directions be zero.

 

[vela]

One mistake I see is the first term in $\dot{x}_\text{cm}$. You're supposed to differentiate with respect to $t$, not $x$.