Physics - EMI/AC Question with Solution | TestHub

For a forced harmonic oscillator, the displacement is given by $x(t)=A\cos (\omega t-\varphi )$, where $\omega$ is the driving frequency and $\varphi$ is the phase difference between the driving force and the displacement.

The phase angle $\varphi$ is given by $\tan \varphi =\frac{b\omega }{m({\omega }_0^2-{\omega }^2)}$, where 'b' is the damping coefficient, 'm' is the mass, and ${\omega }_0$ is the natural frequency. At resonance, the driving frequency $\omega$ is equal to the natural frequency ${\omega }_0$. Therefore, the denominator $m({\omega }_0^2-{\omega }^2)$ becomes zero. This implies $\tan \varphi$ approaches infinity (assuming $b\omega \ne 0$ for a damped system). Thus, $\varphi =\frac{\pi }{2}$ (or 90 degrees). This means the displacement lags the driving force by $\frac{\pi }{2}$ at resonance.