Question

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

(a) 408.25 Hz; (b) 24th harmonic.

Step-by-step solution

Convert Units

The mass of the wire is given in grams and must be converted to kilograms for our calculations since the standard unit of mass in physics is the kilogram. Therefore, we convert 3.00 g to kg:$$3.00\text{g}=0.003\text{kg}$$

Calculate Linear Mass Density

Next, we determine the linear mass density ($\mu$) of the wire, which is mass per unit length. It's given by the formula:$$\mu =\frac{m}{L}$$Where:- $m=0.003\text{kg}$ (mass of the wire)- $L=0.400\text{m}$ (length of the wire)Substituting the values:$$\mu =\frac{0.003}{0.400}=0.0075\text{kg/m}$$

Calculate Speed of Wave on Wire

The speed of the wave on the string $v$ is calculated using the formula:$$v=\sqrt{\frac{T}{\mu }}$$Where:- $T=800\text{N}$ (tension)- $\mu =0.0075\text{kg/m}$ (linear mass density)$$v=\sqrt{\frac{800}{0.0075}}\approx 326.60\text{m/s}$$

Calculate Fundamental Frequency

The fundamental frequency $f$ of a stretched string is given by the formula:$$f=\frac{v}{2L}$$Substitute the values we have:$$f=\frac{326.60}{2\times 0.400}\approx 408.25\text{Hz}$$

Determine the Highest Audible Harmonic

To find the highest harmonic number $n$ that can be heard ($\le 10,000\text{Hz}$), use the relation:$$f_n=n\times f\le 10,000$$Using the fundamental frequency $f=408.25\text{Hz}$:$$n\le \frac{10,000}{408.25}\approx 24.50$$Since $n$ must be an integer, the highest harmonic is 24.

Key concepts

Harmonics

Harmonics are fascinating phenomena observed with vibrating systems like strings and air columns. They occur when a wave can exist at multiple frequencies that are integer multiples of a fundamental frequency. The fundamental frequency is often the lowest frequency and the first harmonic. On a musical string instrument, this represents the base note you hear, while the higher harmonics contribute to the instrument's timbre, providing richness and depth to the sound.

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

Linear mass density is the mass of a wire or string divided evenly along its length. It's denoted by the Greek letter μ (mu) and has the units of kilograms per meter (kg/m). This measure is crucial when examining how a string vibrates, as it affects both the tension and the wave speed.

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

Wave speed on a string or wire is a pivotal concept when considering vibrations and sound. It can be calculated using the tension in the string and its linear mass density. The formula to determine wave speed is:

• $v=\sqrt{\frac{T}{\mu }}$
• 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.