A particle of mass $m$ is attached to a spring of force constant $k$ and performs simple harmonic motion. If the particle is subjected to a damping force proportional to its velocity with damping constant $b$, and an external driving force $F(t)=F_0\cos (\omega t)$ is applied, what is the phase difference between the displacement and the driving force at resonance?

Solution

1. The equation of motion for a driven damped harmonic oscillator is $m\ddot{x}+b\dot{x}+kx=F_0\cos (\omega t)$.
2. At resonance, the driving frequency $\omega$ matches the natural frequency, causing the driving force to be exactly in phase with the velocity of the particle.
3. Since the displacement $x(t)$ lags behind the velocity $v(t)=\dot{x}(t)$ by a phase of $\pi /2$, the displacement must also lag behind the driving force by $\pi /2$.
4. Hence the answer is (C).