Problem 61 - Physics

Question

One method for measuring the speed of sound uses standing waves. A cylindrical tube is open at both ends, and one end admits sound from a tuning fork. A movable plunger is inserted into the other end at a distance \(L\) from the end of the tube where the tuning fork is. For a fixed frequency, the plunger is moved until the smallest value of \(L\) is measured that allows a standing wave to be formed. (a) When a standing wave is formed in the tube, is there a displacement node or antinode at the end of the tube where the tuning fork is, and is there a displacement node or antinode at the plunger? (b) How is the smallest value of \(L\) related to the wavelength of the sound? Explain your answers. The tuning fork produces a \(485-\mathrm{Hz}\) tone, and the smallest value observed for \(L\) is \(0.264 \mathrm{~m}\). What is the speed of the sound in the gas in the tube?

Step-by-Step Solution

Answer

The speed of sound in the gas is 256.08 m/s.

Step 1: Understand the Standing Wave Setup

When a standing wave is formed in a cylindrical tube open at both ends, positions on the tube end exhibit displacement antinodes because the air is free to oscillate, resulting in maximum displacement. Conversely, at the plunger, a displacement node forms because it restricts movement, leading to minimum displacement.

Step 2: Determine Standing Wave Pattern

For a cylindrical tube open at both ends, the fundamental (smallest) mode of a standing wave forms when half of a wavelength fits inside the tube. Thus, \( L \) at its smallest for a fundamental standing wave relates to the wavelength \( \lambda \) as \( L = \frac{\lambda}{2} \).

Step 3: Apply the Relation Between Wavelength and Frequency

The speed \( v \) of sound is determined by the relationship \( v = f \cdot \lambda \), where \( f \) is the frequency and \( \lambda \) is the wavelength. From the previous step, \( \lambda = 2L \), substituting we have: \( v = f \cdot 2L \).

Step 4: Calculate the Speed of Sound

Insert the given values into the equation: \( f = 485 \, \text{Hz} \) and \( L = 0.264 \, \text{m} \), to find \( \lambda = 2 \times 0.264 = 0.528 \, \text{m} \). Then calculate the speed: \[ v = 485 \, \text{Hz} \times 0.528 \, \text{m} = 256.08 \, \text{m/s} \].

Key Concepts

Speed of Sound

The speed of sound is essentially how fast sound waves travel through a medium, like air. It can vary greatly depending on the medium's properties, such as temperature, pressure, and density. In our specific exercise, we're using a cylindrical tube to determine this speed. By utilizing the properties of standing waves created by a tuning fork, we can calculate the speed of sound. To find this, we use the formula: \( v = f \cdot \lambda \), where \( v \) is the speed of sound, \( f \) is the frequency of the sound, and \( \lambda \) is the wavelength. This quantifies the relationship between how often the wave peaks and troughs occur (frequency) and the distance between them (wavelength). By knowing these details, we can accurately calculate the speed of sound.

Displacement Node

A displacement node in a standing wave is a point where there is no movement. It is a crucial concept in understanding how standing waves form in tubes. At this point, due to the interaction of progressive waves, there is complete destructive interference, hence no displacement of particles. When sound waves from a tuning fork enter a tube with a plunger at one end, the plunger end serves as a displacement node. This is because the movement of air is restricted by the plunger, preventing any oscillation. Recognizing the placement of these nodes helps us understand the wave pattern and behavior within the tube.

Displacement Antinode

In contrast to a node, a displacement antinode is where particles in the medium reach their maximum displacement. When a sound wave enters a tube open at both ends, the ends create regions where energy is not constrained, allowing the maximum oscillation of particles. The end of the tube where the tuning fork is located acts as a displacement antinode. This is where the air particles freely oscillate back and forth, peaking in their motion. Understanding the locations of antinodes is essential in determining the nature and configuration of the standing wave patterns in the medium.

Wavelength Calculation

Wavelength calculation is an essential aspect of understanding standing waves. It helps to determine how energy is distributed in the medium as sound travels through. In a tube open at both ends, the smallest length \( L \) that allows a standing wave corresponds to half a wavelength.

• For the fundamental frequency, the tube length \( L \) satisfies \( L = \frac{\lambda}{2} \).
• Therefore, to find the full wavelength, double the length of the tube, \( \lambda = 2L \).
• This provides a straightforward manner to relate the physical dimensions of the tube to sound wavelength.

Using these calculations in the context of the problem, we find that if \( L = 0.264 \, \mathrm{m} \), then the wavelength \( \lambda = 0.528 \, \mathrm{m} \). With this information, along with the frequency, we can effectively calculate the speed of sound.