How long would an instrument which is open at one end have to be in order to hear the fifth harmonic at 4000 Hz on an open tube? Assume the speed of sound is 340 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.
How long would an instrument which is open at one end have to be in order to hear the fifth harmonic at 4000 Hz on an open tube? Assume the speed of sound is 340 m/s.
![### Physics Question: Harmonics in Open Tubes
**Question:**
How long would an instrument which is open at one end have to be in order to hear the 5th harmonic at 4000 Hz on an open tube? Assume the speed of sound is 340 m/s.
- 0.21 m
- 0.51 m
- 0.44 m
- 0.11 m
**Explanation:**
To solve this problem, we will use the formula for the length of a pipe open at one end for the nth harmonic:
\[
L = \frac{(2n-1)\lambda}{4}
\]
Here, \( n \) is the harmonic number, and \( \lambda \) is the wavelength of the sound.
First, calculate the wavelength \( \lambda \) using the relationship between speed, frequency, and wavelength:
\[
\lambda = \frac{v}{f}
\]
where
\( v = 340 \ \text{m/s} \) (speed of sound) and \( f = 4000 \ \text{Hz} \).
So,
\[
\lambda = \frac{340 \ \text{m/s}}{4000 \ \text{Hz}} = 0.085 \ \text{m}
\]
For the 5th harmonic, \( n = 5 \).
Using the formula for the length:
\[
L = \frac{(2 \times 5 - 1) \times 0.085 \ \text{m}}{4} = \frac{9 \times 0.085 \ \text{m}}{4} = 0.19 \ \text{m}
\]
Hence, the closest answer from the given options is:
- **0.21 m**
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