The equation of motions for the damped oscillator was: \begin{equation} \overset{\bullet \bullet}{x} + \frac{2}{\tau} \overset{\bullet}{x} + w_0^2 x = 0 \end{equation} In the case of an underdamped oscillator, we solved that to obtain a function for $x(t)$: \begin{equation} x(t) = A e^{-t/\tau} \cos \left(\omega_1 t - \phi \right) \label{eq:dampedsolutionphase} \end{equation} where $\omega_1 = \sqrt{\omega_0^2-\frac{1}{\tau^2}}$

Knowing that at $t=0$, the position and velocity are given by $x_0$ and $v_0$ respectively, find the constants $A$ and $\phi$ in terms of the initial conditions and any other relevant parameters

Calculate the natural frequency $f$ for this mass/spring system.