AP Physics questions on Harmonics and Overtones

📚 Quick Study Guide

• 📏 Harmonics & Overtones: These terms describe the resonant frequencies of a vibrating system (string, air column, etc.).
• fundamental frequency ($f_1$) is the lowest resonant frequency.
• 🎵 Harmonics: Integer multiples of the fundamental frequency ($f_n = nf_1$, where $n = 1, 2, 3,...$).
• 💨 Open-Open or Open-Closed Pipes: Open pipes have all harmonics.
• 🌬️ Closed-Closed Pipes: Closed pipes only have odd harmonics ($f_n = nf_1$, where $n = 1, 3, 5,...$).
• 🧵 Strings: For a string fixed at both ends, the wavelength of the $n^{th}$ harmonic is $\lambda_n = \frac{2L}{n}$, where $L$ is the length of the string. The frequency is $f_n = \frac{nv}{2L}$, where $v$ is the wave speed.
• 🔊 Wave Speed: The speed of a wave on a string is given by $v = \sqrt{\frac{T}{\mu}}$, where $T$ is the tension and $\mu$ is the linear mass density.

🧪 Practice Quiz

1. A string of length $L$ is fixed at both ends. What is the wavelength of the third harmonic?
   1. $\frac{L}{3}$
   2. $\frac{2L}{3}$
   3. $L$
   4. $2L$
2. A pipe is open at both ends and has a length of $2m$. If the speed of sound is $343 m/s$, what is the frequency of the second harmonic?
   1. $85.75 Hz$
   2. $171.5 Hz$
   3. $257.25 Hz$
   4. $343 Hz$
3. A pipe is closed at one end and open at the other. Which of the following harmonics are present?
   1. All harmonics
   2. Only even harmonics
   3. Only odd harmonics
   4. No harmonics
4. What happens to the frequency of the fundamental mode of a string if the tension is quadrupled?
   1. It is halved
   2. It remains the same
   3. It is doubled
   4. It is quadrupled
5. A string has a length of $1m$ and a mass of $0.01 kg$. If the tension in the string is $100 N$, what is the speed of the wave on the string?
   1. $10 m/s$
   2. $50 m/s$
   3. $100 m/s$
   4. $200 m/s$
6. The frequency of the third harmonic of a vibrating string is 600 Hz. What is the frequency of the fundamental?
   1. 100 Hz
   2. 200 Hz
   3. 300 Hz
   4. 600 Hz
7. What is the relationship between the wavelength of the fundamental frequency and the length $L$ of a string fixed at both ends?
   1. $\lambda = L$
   2. $\lambda = 2L$
   3. $\lambda = \frac{L}{2}$
   4. $\lambda = \frac{L}{4}$