Yujin is playing on a violin string with a length 32 cm and linear density 1.5 g/cm, that resonates with the first overtone from a 1.7-m long organ pipe with one end closed and the other end open that Jeong is playing, what is the tension in the string? Assume the speed of sound in air is 343 m/s.
Properties of sound
A sound wave is a mechanical wave (or mechanical vibration) that transit through media such as gas (air), liquid (water), and solid (wood).
Quality Of Sound
A sound or a sound wave is defined as the energy produced due to the vibrations of particles in a medium. When any medium produces a disturbance or vibrations, it causes a movement in the air particles which produces sound waves. Molecules in the air vibrate about a certain average position and create compressions and rarefactions. This is called pitch which is defined as the frequency of sound. The frequency is defined as the number of oscillations in pressure per second.
Categories of Sound Wave
People perceive sound in different ways, like a medico student takes sound as vibration produced by objects reaching the human eardrum. A physicist perceives sound as vibration produced by an object, which produces disturbances in nearby air molecules that travel further. Both of them describe it as vibration generated by an object, the difference is one talks about how it is received and other deals with how it travels and propagates across various mediums.
![**Problem Statement:**
Yujin is playing on a violin string with a length of 32 cm and a linear density of 1.5 g/cm, that resonates with the first overtone from a 1.7-m long organ pipe with one end closed and the other end open that Jeong is playing. What is the tension in the string? Assume the speed of sound in the air is 343 m/s.
**Explanation:**
In this problem, we need to determine the tension in the violin string, which resonates with a particular organ pipe note. The steps involve converting units, calculating frequencies for both the string and the pipe, and then using the equations of waves on a string to find the tension.
1. **Unit Conversion and Parameters:**
- Length of violin string, \( L_s = 32 \text{ cm} = 0.32 \text{ m} \)
- Linear density of violin string, \( \mu = 1.5 \text{ g/cm} = 0.0015 \text{ kg/m} \)
- Length of organ pipe, \( L_p = 1.7 \text{ m} \)
- Speed of sound in air, \( v = 343 \text{ m/s} \)
2. **Organ Pipe:**
- For an organ pipe with one end closed and the other open, it supports odd harmonics.
- First overtone (second harmonic) frequency formula:
\[
f_1 = \frac{3v}{4L_p} = \frac{3 \times 343}{4 \times 1.7} \approx 151 \text{ Hz}
\]
3. **Violin String:**
- The string's frequency \( f_s \) matches the frequency of the organ pipe's first overtone.
- Fundamental frequency of a string:
\[
f_s = \frac{1}{2L_s}\sqrt{\frac{T}{\mu}}
\]
- Given \( f_s = 151 \text{ Hz} \), we rearrange the formula to solve for tension \( T \):
\[
151 = \frac{1}{2 \times 0.32} \sqrt{\frac{T}{0.0015}}
\]
\[
151 = \frac{1}{0.64} \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb865a02-7408-4eb8-b9f3-2c56928a70f5%2F89fff3e2-72c8-45b6-b5b5-de0285de04d1%2Fssgiwgr_processed.png&w=3840&q=75)
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