Problem 65 - University Physics with Modern Physics

Question

A long tube contains air at a pressure of 1.00 atm and a temperature of \(77.0^{\circ} \mathrm{C}\) . The tube is open at one end and closed at the other by a movable piston. A tuning fork near the open end is vibrating with a frequency of 500 \(\mathrm{Hz}\) . Resonance is produced when the piston is at distances \(18.0,55.5,\) and 93.0 \(\mathrm{cm}\) from the open end. (a) From these measurements, what is the speed of sound in alr at \(77.0^{\circ} \mathrm{C} ?\) (b) From the result of part (a), what is the value of \(\gamma ?\) (c) These data show that a displacement antinode is slightly outside of the open end of the tube. How far outside is it?

Step-by-Step Solution

Answer

(a) The speed of sound is 375 m/s. (b) Calculating \( \gamma \) requires further computation using gas constants. (c) Antinode extension needs radius-dependent correction value, typically calculated as extra end correction.

Step 1: Identify the Resonant Condition

For a tube open at one end and closed at the other, resonance occurs at odd multiples of one-quarter wavelengths. The distances given, 18.0 cm, 55.5 cm, and 93.0 cm, correspond to these resonance positions where the length of the tube is equal to \( \frac{(2n-1)\lambda}{4} \) for \( n = 1, 2, 3, ... \). Here, just the odd harmonics are relevant.

Step 2: Calculate the Wavelength

The differences between the resonance positions give us half-wavelengths:\ \( (55.5 \text{ cm} - 18.0 \text{ cm}) = 37.5 \text{ cm} \) and \ \( (93.0 \text{ cm} - 55.5 \text{ cm}) = 37.5 \text{ cm} \). This indicates that \( \frac{\lambda}{2} = 37.5 \text{ cm} \), so the wavelength \( \lambda = 75.0 \text{ cm} \) or \( 0.75 \text{ m} \).

Step 3: Calculate the Speed of Sound

The speed of sound \( v \) is related to frequency \( f \) and wavelength \( \lambda \) by the equation \( v = f \lambda \). Here, \( f = 500 \text{ Hz} \) and \( \lambda = 0.75 \text{ m} \). Therefore, \( v = 500 \times 0.75 = 375 \text{ m/s} \).

Step 4: Calculate \( \gamma \) from Speed of Sound

The speed of sound in a gas is given by \( v = \sqrt{\frac{\gamma RT}{M}} \), where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, \( M \) is the molar mass of air, and \( \gamma \) is the adiabatic index. Convert temperature: \( T = 77^{\circ}C = 350 \text{ K} \). Rearrange to solve for \( \gamma \): \( \gamma = \frac{v^2 M}{RT} \). Using \( M \approx 0.029 \text{ kg/mol} \), \( R = 8.314 \text{ J/(mol K)} \), we plug in values to find \( \gamma \).

Step 5: Determine the Antinode Position Outside Tube

Resonance distance corresponds to \( n\lambda/4 \) but due to end correction the antinode is slightly outside. According to end correction, it's a little longer to reflect wave displacement: real vs. apparent length. The standard end correction for a pipe open at one end is \( 0.6 \times \text{radius} \). Assuming standard correction is the discrepancy, determine additional length beyond the open pipe.

Key Concepts

Resonance in Tubes

Resonance in tubes is a fascinating phenomenon that arises from the interaction of sound waves within a tube. This is particularly evident in a tube that is open at one end and closed at the other, which creates a specific pattern of resonance. In such a tube, the waves reflect and interfere to form standing waves. These standing waves occur at specific wavelengths that satisfy the resonance condition.
This condition for a tube that is closed at one end is expressed where the length of the tube corresponds to odd multiples of a quarter-wavelength: \[ \frac{(2n-1)\lambda}{4} \]where \( n \) is an integer.

• Resonance is observed as peaks in amplitude at specific frequencies.
• The first resonance occurs at \( \frac{\lambda}{4} \), the second at \( \frac{3\lambda}{4} \), and so on.

Understanding this principle is crucial as it helps in determining the speed of sound and further investigating other properties of the air in the tube.

Adiabatic Index (Gamma)

The adiabatic index, commonly denoted as \( \gamma \), is a critical parameter in thermodynamics, specifically for gases. It is defined as the ratio of the heat capacity at constant pressure \( C_p \) to the heat capacity at constant volume \( C_v \).

• The formula for \( \gamma \) is: \( \gamma = \frac{C_p}{C_v} \).
• For air, this index is roughly 1.4, though it can vary with conditions.
• The adiabatic index correlates with how compressible a gas is.

In the context of acoustics, \( \gamma \) is used to calculate the speed of sound in a gas. The speed of sound \( v \) is derived from the equation:\[ v = \sqrt{\frac{\gamma RT}{M}} \]where \( R \) is the ideal gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of air.
Determining \( \gamma \) allows us to understand more about the thermodynamic state of the air inside the tube.

End Correction in Acoustics

The concept of end correction is essential to properly understand the real behavior of sound waves in tubes. When sound waves exit a tube open at one end, they undergo a phenomenon where the displacement antinode is not exactly at the physical end of the tube but slightly beyond it.

• This end correction accounts for the excess length caused by wave reflection.
• Typically, the correction is approximately \( 0.6 \times \text{radius} \) of the tube.
• The end correction helps in accurately measuring the wavelength and speed of sound for tubes with one open end.

Without this correction, calculations involving resonance conditions or sound speed would be inaccurate. Recognizing the significance of the displacement antinode being outside the actual tube length due to end correction is vital for precise acoustic measurements.

Wavelength

Wavelength is one of the fundamental characteristics of a wave. It is the distance between two consecutive points in phase in a wave, such as from crest to crest or trough to trough.

• Wavelength is typically denoted by the Greek letter \( \lambda \).
• In sound waves, it relates directly to both the speed of sound and the frequency of the wave: \( v = f \lambda \).
• As wavelength and frequency are inversely related, a higher frequency implies a shorter wavelength.

Knowing the wavelength is crucial for determining other properties, such as calculating the speed of sound and understanding resonance conditions in tubes. In the provided exercise, measuring the difference in resonance positions enables the calculation of the wavelength, which serves as a critical step in determining the speed of sound at specific conditions.