Problem 46 - University Physics with Modern Physics

Question

One string of a certain musical instrument is 75.0 \(\mathrm{cm}\) long and has a mass of 8.75 g. It is being played in a room where the speed of sound is 344 \(\mathrm{m} / \mathrm{s}\) . (a) To what tension must you adjust the string so that, when vibrating in its second overtone, it produces sound of wavelength 0.765 \(\mathrm{m} ?\) (Assume that the breaking stress of the wire is very large and isn't exceeded.) (b) What frequency sound does this string produce in its fundamental mode of vibration?

Step-by-Step Solution

Answer

(a) Adjust the tension to produce a third harmonic frequency of 449.02 Hz; (b) This string produces 149.67 Hz in the fundamental mode.

Step 1: Understand the relationship between overtone and frequency

In musical instruments, the second overtone corresponds to the third harmonic. The wavelength of the sound produced by the second overtone is related to the frequency of vibration by the speed of sound. The formula linking these is given by \[ v = f \lambda \] where \( v \) is the speed of sound, \( f \) is the frequency, and \( \lambda \) is the wavelength.

Step 2: Calculate the frequency of the second overtone

Using the formula \( v = f \lambda \), substitute the given wavelength (\( \lambda = 0.765 \) m) and the speed of sound (\( v = 344 \) m/s) to find the frequency, \( f \), of the second overtone. \[ f = \frac{v}{\lambda} = \frac{344}{0.765} \] Calculate \( f \).

Step 3: Relationship between harmonics

The second overtone (third harmonic) is three times the fundamental frequency (first harmonic), \( f_1 \). Thus, \( f = 3f_1 \). Using the frequency calculated in Step 2, find \( f_1 \). \[ f_1 = \frac{f}{3} \]

Step 4: Determine the wave speed in the string

The speed of the wave on the string, \( v_s \), can be expressed in terms of the frequency of its harmonics and its length \( \frac{1}{2} \lambda \). For the third harmonic, \( v_s = 3f_1 \cdot \lambda_s \), where \( \lambda_s = 2 \cdot \text{Length of string} \). The length of the string is 0.75 m, thus \( \lambda_s = 1.5 \) m.

Step 5: Calculate the tension needed

The wave speed on the string \( v_s \) is also related to the string tension \( T \) and mass per unit length \( \mu \): \[ v_s = \sqrt{\frac{T}{\mu}} \]. First calculate \( \mu = \frac{8.75 g}{0.75 m} \) and then solve for \( T \): \[ T = \mu \cdot v_s^2 \].

Step 6: Calculate the frequency of the fundamental mode

Using the previously calculated \( f_1 \), summarize its value as the frequency produced by the string when vibrating in the fundamental mode.

Key Concepts

Harmonic Frequencies

Harmonic frequencies are essential in understanding how musical instruments produce sound. A harmonic is a whole-number multiple of a fundamental frequency, which is the lowest frequency at which a system naturally resonates. In stringed instruments, harmonics are produced when the string vibrates in segments, creating nodes where the amplitude is zero.

The first harmonic is also known as the fundamental frequency. Entire sections of the string vibrate uniformly, producing the primary musical note. Subsequent harmonics, such as the second or third, produce overtones since the string is divided into smaller vibrating sections. These overtones enhance the sound by adding richness and complexity.

When referring to the 'second overtone,' it is important to understand that it corresponds to the 'third harmonic.' In this situation, the string vibrates in three equal sections, leading to three nodes along the length of the string. This concept is crucial for musicians and physicists alike as it describes how different pitches and tones are produced through varying frequencies.

Wave Speed

Wave speed on a string is a critical element that influences the sound produced by a musical instrument. It represents the speed at which a wave travels along the string. Wave speed can be determined by the tension in the string and the mass per unit length, described by the formula:

\[ v_s = \sqrt{\frac{T}{\mu}} \]Here, \(v_s\) is the wave speed, \(T\) is the tension in the string, and \(\mu\) is the mass per unit length of the string. This relationship shows that increasing the tension or decreasing the mass distribution allows the wave to travel more quickly.

• A higher wave speed results in higher frequencies for the same mode of vibration.
• Consequently, this affects the pitch of the sound produced.

Understanding wave speed is important for tuning instruments, as musicians adjust the tension to achieve the desired pitch.
This knowledge is also used to ensure optimal performance and sound quality.

String Tension

String tension is an integral factor that significantly impacts how a string vibrates, influencing both the frequency and amplitude of the produced sound. Tension refers to the force applied along the string, maintaining its state between the fixed points of the instrument.

In practice, when you increase the tension of a string by tightening it, the frequency of its vibrations also increases. This is conveyed through the formula relating tension, wave speed, and mass per unit length: \[ v_s = \sqrt{\frac{T}{\mu}} \] Given this, greater tension results in a higher wave speed, which in turn increases the frequency produced by the string for a specific harmonic.

• Highly tense strings produce higher-pitched notes.
• Low tension results in deeper, lower-pitched sounds.

Maintaining the right amount of tension ensures that the instrument is in tune. Moreover, understanding how tension relates to wave mechanics helps in customizing the instrument for different musical styles and performing conditions.

Fundamental Frequency

The fundamental frequency is the lowest natural frequency of vibration of a string or any resonant system. It serves as the primary frequency that establishes the base pitch of a musical note. For a string fixed at both ends, this frequency can be calculated as:\[ f_1 = \frac{v_s}{2L} \]where \(f_1\) is the fundamental frequency, \(v_s\) is the wave speed on the string, and \(L\) is the string's length.
The fundamental frequency produces the first harmonic, and all other harmonics are multiples of this frequency. In music, this is what we primarily hear as the note being played. Understanding this concept helps musicians and engineers design and tune instruments accurately.

• For the string length of a musical instrument, adjusting the tension and altering the mass per unit length can fine-tune the fundamental frequency.
• The complexity and richness of the resulting sound depend significantly on how this frequency interacts with its harmonics and overtones.

The recognition of the fundamental frequency provides a basis for the tonal qualities and the sound character of the instrument.