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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 $\tan(\varphi )=\omega \gamma / (\omega_0^2 - \omega^2)$.

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

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?

For a driven damped oscillation, if the driven force $F=F_{ 0 }cos(\omega t )$, then the solution to the motion is $$x=Acos(\omega t+\varphi )$$.

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

The solution to $\varphi$ is $tan(\varphi )=\frac { \omega \gamma }{ { { \omega }_{ 0 } }^{ 2 }-{ \omega }^{ 2 } }$.

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

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 $\tan(\varphi )=\omega \gamma / (\omega_0^2 - \omega^2)$.

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

For a driven damped oscillation, if the driven force $F=F_{ 0 }cos(\omega t )$, then the solution to the motion is $x=Acos(\omega t+\varphi )$. Why the oscillation and the driven force must have the same frequency but can be out of phase? The solution to $\varphi$ is $tan(\varphi )=\frac { \omega \gamma }{ { { \omega }_{ 0 } }^{ 2 }-{ \omega }^{ 2 } }$. Why would $\varphi$ goes to zero(F and x in phase) if $\omega$ goes to zero?

For a driven damped oscillation, if the driven force $F=F_{ 0 }cos(\omega t )$, then the solution to the motion is $$x=Acos(\omega t+\varphi )$$.

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

The solution to $\varphi$ is $tan(\varphi )=\frac { \omega \gamma }{ { { \omega }_{ 0 } }^{ 2 }-{ \omega }^{ 2 } }$.

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