For a driven damped oscillation, if the driven force $F = F_0 \cos(\omega t)$, then the solution to the motion is $$x = A \cos(\omega t+\varphi ) \, .$$

• Why must the the oscillation and the driven force have the same frequency but can be out of phase?

The solution to $\varphi$ is $\tan(\varphi )=\omega \gamma / (\omega_0^2 - \omega^2)$. （$\gamma$ is the damping coefficient divided by mass for a spring system)

• Why would $\varphi$ goes to zero (meaning $F$ and $x$ are in phase) if $\omega$ goes to zero?