Coupled Oscillators

Use the formula for the cosine of a sum to give an even more general formula for simple harmonic motion.

Find the eigenvectors associated with the eigenvalues.

Find the values of $\omega$ in the case where $x(t)=A_{1}\cos(\omega t+\phi)$ and $\theta(t)=A_{2}\cos(\omega t+(\phi+\pi))$.

Build a Wilberforce pendulum out of a Slinky Jr and a soup can. You can drill holes in the can for bolts, either two or four of them, that you can use to adjust the moment of inertia of the can so the periods of vertical and rotational motion coincide.