AP Physics C Questions on Damped Harmonic Motion with Solutions

📚 Quick Study Guide

• 🌊 Damping Force: A force that opposes the motion of an oscillator, causing the amplitude to decrease over time. Often modeled as $F_d = -bv$, where $b$ is the damping coefficient and $v$ is the velocity.
• 📉 Underdamped: The system oscillates with a gradually decreasing amplitude. Occurs when $b < 2\sqrt{mk}$, where $m$ is mass and $k$ is the spring constant.
• 🛑 Critically Damped: The system returns to equilibrium as quickly as possible without oscillating. Occurs when $b = 2\sqrt{mk}$.
• 💀 Overdamped: The system returns to equilibrium slowly without oscillating. Occurs when $b > 2\sqrt{mk}$.
• 🕰️ Angular Frequency (Damped): In an underdamped system, the angular frequency is given by $\omega' = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}$.
• ⚡ Energy Loss: Damping causes energy to dissipate from the system, usually as heat. The rate of energy loss is proportional to the square of the velocity.
• 📏 Amplitude Decay: In an underdamped system, the amplitude decreases exponentially with time, $A(t) = A_0e^{-\frac{b}{2m}t}$, where $A_0$ is the initial amplitude.

🧪 Practice Quiz

1. What type of damping results in the quickest return to equilibrium *without* oscillation?
   1. A) Underdamped
   2. B) Critically Damped
   3. C) Overdamped
   4. D) Undamped
2. A mass-spring system has mass $m = 2 \text{ kg}$ and spring constant $k = 8 \text{ N/m}$. What damping coefficient $b$ would result in critical damping?
   1. A) $2 \text{ kg/s}$
   2. B) $4 \text{ kg/s}$
   3. C) $8 \text{ kg/s}$
   4. D) $16 \text{ kg/s}$
3. In an underdamped system, what happens to the amplitude of oscillation over time?
   1. A) Increases linearly
   2. B) Decreases linearly
   3. C) Increases exponentially
   4. D) Decreases exponentially
4. Which of the following is the correct formula for the damped angular frequency $\omega'$ in an underdamped system?
   1. A) $\omega' = \sqrt{\frac{k}{m} + \frac{b^2}{4m^2}}$
   2. B) $\omega' = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}$
   3. C) $\omega' = \frac{k}{m} - \frac{b}{2m}$
   4. D) $\omega' = \frac{k}{m} + \frac{b}{2m}$
5. What is the primary mechanism by which energy is lost in a damped harmonic oscillator?
   1. A) Gravitational potential energy
   2. B) Kinetic energy
   3. C) Heat dissipation
   4. D) Elastic potential energy
6. If the damping coefficient $b$ is significantly larger than $2\sqrt{mk}$, the system is:
   1. A) Underdamped
   2. B) Critically Damped
   3. C) Overdamped
   4. D) Undamped
7. A damped oscillator has an initial amplitude of $10 \text{ cm}$. After a long time, the amplitude is practically zero. What happened to the energy of the system?
   1. A) It was converted into potential energy.
   2. B) It was converted into kinetic energy.
   3. C) It was dissipated as heat.
   4. D) It remains stored in the spring.