Solved Examples of Driven Oscillations Resonance

📚 Quick Study Guide

• 🎢 Driven Oscillations: Occur when an external periodic force is applied to an oscillating system.
• ⏱️ Driving Frequency ($\omega$): The frequency at which the external force is applied.
• 📉 Damping: Resistance that dissipates energy in an oscillating system, reducing amplitude over time.
• 🎯 Resonance: Occurs when the driving frequency is near the natural frequency ($\omega_0$) of the system, leading to a large amplitude.
• 📐 Natural Frequency ($\omega_0$): The frequency at which a system oscillates freely without any external force. For a simple harmonic oscillator, $\omega_0 = \sqrt{\frac{k}{m}}$, where $k$ is the spring constant and $m$ is the mass.
• ⚡ Amplitude Equation: The amplitude $A$ of a driven oscillator is given by: $A = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (b\omega/m)^2}}$, where $F_0$ is the amplitude of the driving force, $m$ is the mass, $b$ is the damping coefficient, $\omega_0$ is the natural frequency, and $\omega$ is the driving frequency.
• 📊 Resonance Peak: The amplitude is maximum when $\omega$ is close to $\omega_0$. The sharpness of the peak depends on the damping; less damping means a sharper peak.

Practice Quiz

1. A mass-spring system is driven by an external force. Resonance occurs when:
   1. The driving frequency is much greater than the natural frequency.
   2. The driving frequency is much smaller than the natural frequency.
   3. The driving frequency is equal to the natural frequency.
   4. The driving frequency is zero.
2. What effect does increasing the damping coefficient have on the resonance peak of a driven oscillator?
   1. It increases the height and narrows the width.
   2. It decreases the height and broadens the width.
   3. It increases the height and broadens the width.
   4. It decreases the height and narrows the width.
3. A driven oscillator has a natural frequency of 5 Hz. Which driving frequency will result in the largest amplitude if damping is minimal?
   1. 1 Hz
   2. 3 Hz
   3. 5 Hz
   4. 7 Hz
4. The amplitude of a driven oscillator at resonance is inversely proportional to:
   1. The driving frequency.
   2. The natural frequency.
   3. The damping coefficient.
   4. The mass of the oscillator.
5. What happens to the phase difference between the driving force and the oscillator's displacement at resonance?
   1. It is 0 degrees.
   2. It is 45 degrees.
   3. It is 90 degrees.
   4. It is 180 degrees.
6. If the driving frequency is much higher than the natural frequency, what is the approximate phase difference between the driving force and the oscillator's displacement?
   1. 0 degrees
   2. 90 degrees
   3. 180 degrees
   4. 270 degrees
7. A simple pendulum is driven by a periodic force. To achieve resonance, you should:
   1. Apply a constant force.
   2. Apply a force at a frequency much higher than its natural frequency.
   3. Apply a force at a frequency much lower than its natural frequency.
   4. Apply a force at a frequency close to its natural frequency.