Problem 15 - Physics

Question

A \(3.00-\mathrm{kHz}\) tone is being produced by a speaker with a diameter of \(0.175 \mathrm{~m} .\) The air temperature changes from 0 to \(29^{\circ} \mathrm{C}\). Assuming air to be an ideal gas, find the change in the diffraction angle \(\theta\).

Step-by-Step Solution

Answer

The change in diffraction angle \( \Delta \theta \) is found by calculating \( \theta_2 - \theta_1 \).

Step 1: Calculate Initial Speed of Sound

The speed of sound in air can be calculated using the formula \( v = \sqrt{\gamma \cdot R \cdot T/M} \) where \(\gamma\) is the adiabatic index (1.40 for air), \(R\) is the universal gas constant (8.314 J/(mol K)), \(M\) is the molar mass of air (0.029 kg/mol), and \(T\) is the temperature in Kelvin. Initially, \( T_1 = 0 \degree C = 273.15 K \). Therefore, the initial speed of sound \( v_1 \) is calculated.

Step 2: Calculate Final Speed of Sound

Similarly, calculate the speed of sound at the final temperature. At \( T_2 = 29 \degree C = 302.15 K \). Use the same formula as in Step 1 to find the final speed of sound \( v_2 \).

Step 3: Find Wavelengths

The wavelength \( \lambda \) of the sound is given by \( \lambda = \frac{v}{f} \) where \( f = 3000 \text{ Hz} \) is the frequency. Calculate the initial wavelength \( \lambda_1 = \frac{v_1}{3000} \) and the final wavelength \( \lambda_2 = \frac{v_2}{3000} \).

Step 4: Calculate Diffraction Angle

The diffraction angle \( \theta \) is calculated using the formula \( \theta = \arcsin\left( \frac{\lambda}{d} \right) \) where \( d = 0.175 \text{ m} \) is the diameter of the speaker. Compute the initial diffraction angle \( \theta_1 = \arcsin\left( \frac{\lambda_1}{d} \right) \) and the final diffraction angle \( \theta_2 = \arcsin\left( \frac{\lambda_2}{d} \right) \).

Step 5: Calculate Change in Diffraction Angle

The change in diffraction angle is given by \( \Delta \theta = \theta_2 - \theta_1 \). Subtract the initial diffraction angle from the final diffraction angle to find \( \Delta \theta \).

Key Concepts

Speed of Sound

The speed of sound in air is not a constant value. It varies based on several factors, mainly the temperature of the air. When sound travels, it moves as a pressure wave through the air molecules. These molecules collide with each other, transferring the sound wave forward.
A fundamental formula to determine the speed of sound is \[ v = \sqrt{\gamma \cdot R \cdot T/M} \]
where:

• \( \gamma \) is the adiabatic index, typically 1.40 for air.
• \( R \) is the universal gas constant, 8.314 J/(mol K).
• \( M \) is the molar mass of air, 0.029 kg/mol.
• \( T \) is the temperature in Kelvin.

The speed of sound increases with temperature. For example, when the temperature is warmer, air molecules move faster, allowing sound waves to propagate more quickly. This is why in the original problem, when the temperature changes from 0°C to 29°C, the speed of sound increases.

Wavelength Calculation

The wavelength of a sound wave is a crucial property that relates to how we perceive sound pitch. It is calculated using the formula:\[ \lambda = \frac{v}{f} \]
where:

• \( \lambda \) is the wavelength (in meters).
• \( v \) is the speed of sound (in meters per second).
• \( f \) is the frequency of the sound (in Hertz).

In this exercise, the sound frequency is given as 3000 Hz (3 kHz). With the initial and final speeds of sound calculated based on initial and final temperatures, we can compute the corresponding wavelengths.
Wavelength changes because when the speed of sound increases (due to a rise in temperature), for a given frequency, the wavelength also lengthens. Conversely, when the temperature is low, the speed of sound is slower, leading to a shorter wavelength.

Ideal Gas Assumption

Air is often modeled as an ideal gas to simplify calculations. The ideal gas law states that \[ PV = nRT \]
describes the relationship between pressure \( P \), volume \( V \), and temperature \( T \) of a gas, assuming a number of moles \( n \) and the gas constant \( R \).
The significance of assuming air to be an ideal gas lies in its mathematical simplicity, allowing us to use formulas that relate gas properties straightforwardly. This is crucial for calculating the speed of sound because it relies on this theoretical simplification.
Despite real-world deviations due to various factors like humidity or high pressure, the ideal gas assumption generally provides adequate precision for most basic phenomena, such as changes in sound speed with temperature in everyday environments.

Temperature Effects

Temperature profoundly impacts several acoustical properties. When temperature rises, the energy of air molecules increases, and they vibrate more vigorously, which affects sound propagation.
Some key effects of temperature on sound include:

• **Change in Speed:** As previously mentioned, higher temperatures lead to higher speeds of sound.
• **Frequency Stability:** The frequency of sound, in general, doesn't change with temperature, since it is determined by the source like a speaker.
• **Wavelength Variation:** As the sound speed changes while frequency remains the same, the wavelength varies directly with temperature.

Understanding temperature effects is crucial for calculating diffraction angles, as in the given exercise. The change in diffraction angle observed when temperature alters is primarily due to the change in speed of sound, which leads to a different wavelength.