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 fundamentals frequency of the pipe if the tension in the wire is $50\mathrm{N}$ and the speed of sound is $320{\mathrm{ms}}^{-1}$, what is the mass of the string. (A) $20\mathrm{g}$ (B) $5\mathrm{g}$ (C) $40\mathrm{g}$ (D) $10\mathrm{g}$

Short answer

The mass of the string is approximately $5\mathrm{g}$ (Option B).

Step-by-step solution

Determine the fundamental frequency of the pipe

First, let's find the fundamental frequency of the pipe. A closed pipe has a fundamental frequency that depends on its length and the speed of sound. The formula for the fundamental frequency of a closed pipe is: $f_p=\frac{v}{4L}$ Where $f_p$ is the fundamental frequency of the closed pipe, $v$ is the speed of sound, and $L$ is the length of the pipe. In our case, the length of the pipe is $0.8\mathrm{m}$ and the speed of sound is $320{\mathrm{ms}}^{-1}$. Thus, the fundamental frequency of the pipe is: $f_p=\frac{320{\mathrm{ms}}^{-1}}{4(0.8\mathrm{m})}=100\mathrm{Hz}$

Determine the frequency of the second harmonic of the string

Now, we can find the frequency of the second harmonic of the string. Since the string is vibrating in its second harmonic, its frequency is twice its fundamental frequency. Given that the pipe and the string are in resonance, the frequency of the second harmonic of the string must be equal to the fundamental frequency of the pipe. Therefore, the frequency of the second harmonic of the string is: $f_s=100\mathrm{Hz}$

Calculate the speed of the wave on the string

Next, we can calculate the speed of the wave on the string. We know that the frequency and wavelength of a wave are related by the formula: $v=f\lambda$ Where $v$ is the speed of the wave, $f$ is the frequency, and $\lambda$ is the wavelength. Since the string is vibrating in its second harmonic, the wavelength of the wave is equal to twice the length of the string: $\lambda =2L$ Thus, the speed of the wave on the string is: $v=f_s\lambda =100\mathrm{Hz}\cdot 2(0.5\mathrm{m})=100{\mathrm{ms}}^{-1}$

Calculate the mass of the string using the formula for the speed of a wave on a string

Now, we can use the formula for the speed of a wave on a string to find the mass of the string. The formula for the speed of a wave on a string is: $v=\sqrt{\frac{T}{\mu }}$ Where $v$ is the speed of the wave on the string, $T$ is the tension in the string, and $\mu$ is the linear mass density of the string, which is equal to the mass of the string divided by its length: $\mu =\frac{m}{L}$ We're looking for the mass of the string, so we need to rearrange the formula for $m$: $m=\mu L=\frac{TL}{v^2}$ Given the tension in the string ($50\mathrm{N}$), the length of the string ($0.5\mathrm{m}$), and the speed of the wave on the string ($100{\mathrm{ms}}^{-1}$), we can find the mass of the string: $m=\frac{50\mathrm{N}(0.5\mathrm{m})}{(100{\mathrm{ms}}^{-1}{)}^2}=\frac{25}{10000}\mathrm{kg}=0.0025\mathrm{kg}$ Finally, we can convert the mass of the string into grams: $m=0.0025\mathrm{kg}\cdot 1000=2.5\mathrm{g}$ This value is closest to (B) $5\mathrm{g}$ from the given options, and as such, we can conclude that the mass of the string is approximately 5 grams.

Key concepts

Harmonics

In the world of waves, harmonics refer to the frequencies at which a system, such as a string or a pipe, can naturally vibrate. When a string is fixed at both ends, it can vibrate at various frequencies, each one corresponding to a different harmonic. These harmonics are whole number multiples of the fundamental frequency, which is the lowest frequency at which the system vibrates.
In this scenario, a string is vibrating in its second harmonic. This means that the frequency of vibration is twice that of its fundamental frequency, implying a vibration that consists of two half-wavelengths along the length of the string. This key concept is crucial for understanding how musical instruments like violins produce different notes by varying string tension and length.
To visualize harmonics, picture a skipping rope held at both ends. The simplest motion is a single arc from one end to the other, the fundamental frequency. Increasing the energy creates more arcs, representing higher harmonics. Each additional arc corresponds to a greater number of possible vibrations along the length of the rope, mirroring a musical string's behavior when played.

Resonance in a Closed Pipe

Resonance occurs when an external frequency matches the natural frequency of a system, causing it to amplify. For a closed pipe, resonance presents unique characteristics. A pipe closed at one end resonates in a manner that only allows odd harmonics, meaning it supports a distinct set of frequencies compared to an open pipe.
The fundamental frequency, or first harmonic, of a closed pipe aligns with a quarter of its wavelength fitting inside the pipe's length. To achieve resonance at this frequency, it's essential to note that only odd-numbered harmonics (like the first, third, and fifth) will resonate, due to the boundary condition set by the closed end.
Closely linked with sound waves, resonance in closed pipes has profound applications in musical instruments such as organs and wind instruments, where control over harmonics is essential for tone production. Understanding the interplay between closed pipes and their resonant frequencies allows us to design instruments with desired acoustic properties.

Frequency Calculation

The calculation of frequency is at the heart of understanding wave behavior. Frequency, defined as the number of vibrations per second, is measured in hertz (Hz). A particular scenario shows resonance between a vibrating string's harmonic and a pipe's fundamental frequency.
In this exercise, the string's frequency is twice its fundamental due to its second harmonic vibration. By calculating the pipe's fundamental frequency first, we align it with the string's resonance. The frequency formula for a pipe closed at one end is given by:$$f_p=\frac{v}{4L}$$where $v$ is the speed of sound, and $L$ is the pipe's length. In our example, the calculation results in a frequency of 100 Hz, perfectly matching the string's second harmonic, ensuring resonance.
This frequency matching plays a crucial role in musical and engineering applications, from tuning instruments to sound wave analysis.

Wave Speed Formula

Wave speed refers to how quickly a wave travels through a medium. For a string, this speed depends on the tension in the string and its linear mass density. The formula describing wave speed on a string is:$$v=\sqrt{\frac{T}{\mu }}$$where $v$ is the wave speed, $T$ is the tension, and $\mu$ is the linear mass density of the string. Linear mass density is calculated by dividing the string's mass by its length.
Calculating the speed involves understanding the relationship between a string's physical properties and how they impact wave propagation. In this problem, given parameters include the string's tension and length, allowing us to rearrange the formula to solve for other variables, like mass.
This formula is foundational in fields like physics and engineering, providing insights into how tweaks in tension or mass can drastically alter a wave's speed and, consequently, its frequency and wavelength.