Chapter 15: Problem 40
A piano tuner stretches a steel piano wire with a tension of 800 N. The steel wire is 0.400 m long and has a mass of 3.00 g. (a) What is the frequency of its fundamental mode of vibration? (b) What is the number of the highest harmonic that could be heard by a person who is capable of hearing frequencies up to 10,000 Hz?
Short Answer
Step by step solution
Convert Units
Calculate Linear Mass Density
Calculate Speed of Wave on Wire
Calculate Fundamental Frequency
Determine the Highest Audible Harmonic
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.
Harmonics
Understanding harmonics is key because it helps us calculate higher frequencies that a vibrating object can produce. For a wire or string, each harmonic increases in frequency by whole number multiples. If the wire's fundamental frequency is 408.25 Hz, the next harmonic (the second harmonic) would be twice this frequency, and so on. This is particularly useful in music to help understand notes and their octaves. Learn about harmonics, and you'll better appreciate how musicians can produce a range of notes with just a few strings or instruments.
Linear Mass Density
In the example exercise, we calculated the linear mass density of a piano wire by dividing the total mass (0.003 kg) by its length (0.400 m), resulting in 0.0075 kg/m. This density is important because it directly influences the string's wave speed and, subsequently, the frequencies of the harmonics it can produce. In musical terms, the linear mass density affects which notes are heard and their pitch. Thicker or denser strings typically have lower pitches, while lighter, less dense strings tend to have higher pitches.
Wave Speed
- \( v = \sqrt{\frac{T}{\mu}} \)
- where \( v \) is the wave speed, \( T \) is the string tension, and \( \mu \) is the linear mass density.
In our exercise, with a tension of 800 N and a linear mass density of 0.0075 kg/m, we computed the wave speed as approximately 326.60 m/s. This speed guides how quickly waves or vibrations travel along the string. Faster wave speeds lead to higher frequency sounds. Understanding wave speed is essential for tuning string instruments, designing musical instruments, and any applications involving waves and vibrations, including engineering and multimedia fields. Wave speed, in the context of sound, influences how far and how fast sound can travel, impacting both musical acoustics and communication systems.