Chapter 16: Problem 32
You have a stopped pipe of adjustable length close to a taut 62.0-cm, 7.25-g wire under a tension of 4110 N. You want to adjust the length of the pipe so that, when it produces sound at its fundamental frequency, this sound causes the wire to vibrate in its second \(overtone\) with very large amplitude. How long should the pipe be?
Short Answer
Step by step solution
Calculate the Mass Per Unit Length of the Wire
Determine the Fundamental Frequency of the Wire
Calculate the Frequency of the Second Overtone
Match Pipe's Fundamental Frequency with Wire's Second Overtone
Solve for the Pipe Length
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Stopped Pipe
In a stopped pipe, the fundamental frequency occurs when there is a quarter-wave within the pipe. This sets the foundation for harmonic frequencies as subsequent overtones are formed by fitting additional quarter-waves inside the same pipe.
These resonate at odd multiples of the fundamental frequency. This unique property makes stopped pipes integral to various musical instruments like clarinets and organs.
Harmonic Frequencies
For stopped pipes, the harmonics occur at odd multiples of the fundamental frequency. Hence, the first harmonic (or fundamental frequency) equals the speed of sound divided by 4 times the length of the pipe.
The second harmonic frequency, in this case, is three times the fundamental frequency, showcasing how specific structures limit the types of harmonics that can manifest.
Fundamental Frequency
- \( f \) is the fundamental frequency,
- \( v \) is the speed of sound,
- \( L \) is the length of the pipe.
Second Overtone
To find the frequency for the second overtone, we multiply the fundamental frequency by three (for a vibrating string, this factor depends on the type of pipe or string and how its modes are set).
The second overtone's frequency is pivotal in achieving resonance conditions and is crucial for the analysis of sound systems and musical acoustics.
Vibrating String
The formula for the fundamental frequency of a vibrating string is given by:\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]Where:
- \( f \) is the frequency,
- \( L \) is the length of the string,
- \( T \) is the tension,
- \( \mu \) is the mass per unit length.
Speed of Sound
Typically, the speed of sound in air is around 343 m/s under standard temperature and pressure conditions.
This speed can vary based on atmospheric conditions such as temperature and pressure. In many exercises involving acoustic resonators, this constant helps calculate the length of pipes or strings necessary to produce a particular frequency.