Question

A hollow pipe of length \(0.8 \mathrm{~m}\) is closed at one end. At its open end a \(0.5 \mathrm{~m}\) long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the wire is \(50 \mathrm{~N}\) and the speed of sound is \(320 \mathrm{~ms}^{-1}\), the mass of the string is A) 5 grams B) 10 grams C) 20 grams D) 40 grams

Short answer

The mass of the string is 10 grams (Option B).

Step-by-step solution

Calculate the fundamental frequency of the pipe

The fundamental frequency (first harmonic) for a pipe closed at one end is given by the formula: \( f_1 = \frac{v}{4L} \), where \( v \) is the speed of sound and \( L \) is the length of the pipe. Substitute the given values (\( v = 320 \mathrm{ms}^{-1} \) and \( L = 0.8 \mathrm{m} \)) to find the fundamental frequency.

Calculate the frequency of the vibrating string

The string is vibrating in its second harmonic (first overtone), so its frequency will be twice the fundamental frequency. The formula for the fundamental frequency of a string is: \( f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \), where \( L \) is the length of the string, \( T \) is the tension and \( \mu \) is the mass per unit length. The frequency of the second harmonic will be \( f_2 = 2f \).

Determine the mass per unit length of the string

The resonance condition states that the frequency of the pipe's fundamental is equal to the frequency of the second harmonic of the string. By equating the two frequencies, \( f_1 = f_2 \), solve for \( \mu \), which is the mass per unit length of the string.

Calculate the total mass of the string

With the mass per unit length \( \mu \) determined, the total mass \( m \) of the string can be calculated using the formula: \( m = \mu L \), where \( L \) is the length of the string. Substitute the computed value of \( \mu \) and the string's length \( L = 0.5 \mathrm{m} \) to obtain the mass of the string.

Key concepts

Fundamental Frequency

The fundamental frequency, also known as the first harmonic, is the lowest frequency at which an object can naturally vibrate. In the context of musical instruments, such as pipes and strings, this frequency determines the pitch of the note that is heard when the object is set into vibration.

For a pipe that is closed at one end, like in the exercise, the fundamental frequency is calculated using the formula:
\[ f_1 = \frac{v}{4L} \]
where \( v \) represents the speed of sound through the air inside the pipe and \( L \) is the length of the pipe. The end that is closed reflects the wave causing the air column inside the pipe to resonate at a specific frequency. This principle is what allows musical instruments such as flutes and organs to produce distinct notes.

Harmonics

Harmonics, or overtones, are higher frequencies at which an object can also naturally vibrate. These frequencies are multiples of the fundamental frequency and contribute to the richness and color of the sound produced. In the case of the string in the exercise, it vibrates in its second harmonic, which is the first overtone and is twice the frequency of the fundamental.

Mathematically, the frequency of the \(n\)th harmonic of a string is given by:
\[ f_n = n \cdot \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
where \( n \) is the harmonic number, \( L \) is the length of the string, \( T \) is the tension applied, and \( \mu \) is the mass per unit length. For the second harmonic, we would set \( n = 2 \) in this formula. The occurrence of harmonics is critical in music and acoustics, as they influence the timbre or quality of the sound we hear.

Mass Per Unit Length of a String

The mass per unit length of a string, denoted as \( \mu \), is a key property in understanding the dynamics of vibrating strings and plays a vital role in the sonic characteristics they produce. It is calculated by dividing the total mass of the string by its length. This value directly affects how different frequencies, including the fundamental frequency and harmonics, resonate on the string.

Mathematically, the mass per unit length appears in the formula:
\[ \mu = \frac{T}{(2L f)^2} \]
where \( T \) represents the tension in the string, \( L \) is the length of the string, and \( f \) is the frequency at which the string vibrates. Lower mass per unit length usually allows for higher vibrational frequencies, whereas a string with higher mass per unit length vibrates at lower frequencies. This concept is crucial when designing musical instruments and understanding the physics of waves and vibrations.

Speed of Sound

The speed of sound is the rate at which sound waves travel through a medium. It varies depending on factors such as the medium's density, temperature, and state of matter. In air, at room temperature, the speed of sound is approximately 343 meters per second (m/s). In our exercise, the speed of sound is given as \( 320 \mathrm{ms}^{-1} \).

The speed of sound in air is critical for calculating the fundamental frequency of the hollow pipe. This value is used in conjunction with the length of the pipe to determine the resonant frequencies, including the fundamental frequency. Understanding the speed of sound is essential in various applications, ranging from musical instruments to technological devices like sonar and echolocation.