Solved examples of forced oscillations with damping

📚 Quick Study Guide

• 🔍 Forced oscillation occurs when an external periodic force acts on a damped oscillator.
• 🍎 The equation of motion is given by: $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = F_0\cos(\omega t)$, where $m$ is mass, $b$ is the damping coefficient, $k$ is the spring constant, $F_0$ is the amplitude of the driving force, and $\omega$ is the driving frequency.
• 💡 The steady-state solution has the form: $x(t) = A\cos(\omega t - \phi)$, where $A$ is the amplitude and $\phi$ is the phase lag.
• 📐 The amplitude $A$ is given by: $A = \frac{F_0}{\sqrt{(k - m\omega^2)^2 + (b\omega)^2}}$.
• ⏱️ The phase lag $\phi$ is given by: $\tan(\phi) = \frac{b\omega}{k - m\omega^2}$.
• 🔥 Resonance occurs when the driving frequency $\omega$ is close to the natural frequency $\omega_0 = \sqrt{\frac{k}{m}}$. The amplitude is maximum at resonance.
• 📊 Damping reduces the amplitude of oscillations and broadens the resonance peak.

Practice Quiz

1. What is the equation of motion for a forced oscillation with damping?
   1. $m\frac{d^2x}{dt^2} + kx = 0$
   2. $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = F_0\cos(\omega t)$
   3. $m\frac{d^2x}{dt^2} = F_0\cos(\omega t)$
   4. $b\frac{dx}{dt} + kx = 0$
2. The amplitude $A$ of a forced oscillation with damping is given by which formula?
   1. $A = \frac{F_0}{\sqrt{(k - m\omega^2)^2 + (b\omega)^2}}$
   2. $A = \frac{F_0}{\sqrt{(k + m\omega^2)^2 - (b\omega)^2}}$
   3. $A = \frac{F_0}{(k - m\omega^2)^2 + (b\omega)^2}$
   4. $A = \frac{F_0}{\sqrt{(k - m\omega)^2 + (b\omega)^2}}$
3. What does the phase lag $\phi$ represent in forced oscillations?
   1. The time delay between the driving force and the oscillator's response.
   2. The maximum displacement of the oscillator.
   3. The frequency of the driving force.
   4. The damping coefficient.
4. The phase lag $\phi$ is given by which formula?
   1. $\tan(\phi) = \frac{b\omega}{k - m\omega^2}$
   2. $\sin(\phi) = \frac{b\omega}{k - m\omega^2}$
   3. $\cos(\phi) = \frac{b\omega}{k - m\omega^2}$
   4. $\phi = \frac{b\omega}{k - m\omega^2}$
5. Under what condition does resonance occur in forced oscillations?
   1. When the driving frequency is much smaller than the natural frequency.
   2. When the driving frequency is equal to the natural frequency.
   3. When the driving frequency is much larger than the natural frequency.
   4. Resonance never occurs in damped oscillations.
6. What effect does damping have on the amplitude of oscillations?
   1. Damping increases the amplitude.
   2. Damping decreases the amplitude.
   3. Damping has no effect on the amplitude.
   4. Damping only affects the frequency.
7. How does damping affect the resonance peak?
   1. Damping sharpens the resonance peak.
   2. Damping broadens the resonance peak.
   3. Damping shifts the resonance peak to higher frequencies.
   4. Damping has no effect on the resonance peak.