91Ó°ÊÓ

The speed of sound in a medium is influenced by factors such as temperature and pressure. In this exercise, sound travels in a tube filled with air at 77.0\( ^\circ \)C and 1.00 atm pressure. To find the speed of sound, we use the equation \[ v = f \lambda \]where \( f \) is the frequency of the sound, and \( \lambda \) is the wavelength.
With a tuning fork frequency of 500 Hz and a calculated wavelength of 0.75 m, the speed of sound in the tube is:

• Multiply the frequency by the wavelength: \[ v = 500 \space \text{Hz} \times 0.75 \space \text{m} = 375 \space \text{m/s} \]

This value reflects how fast sound waves travel through the air under the given conditions. Typically, warm air supports faster sound wave propagation due to increased molecular movement.

In wave mechanics, understanding the relationship between the length of a wave and its frequency is crucial. In the given exercise, the resonance condition influences the calculation of the wavelength in the tube. When sound resonates, it means that there is a standing wave pattern:
Successive lengths where resonance occurs can tell us about the wavelength. With the closed tube, resonance length differences relate to half wavelengths.

• Identify half-wavelength difference: \[ \Delta L = 55.5 \space \text{cm} - 18.0 \space \text{cm} = 37.5 \space \text{cm} \]
• \( \Delta L = 93.0 \space \text{cm} - 55.5 \space \text{cm} = 37.5 \space \text{cm} \)
• These calculations confirm that half of the wavelength is 37.5 cm, thus:
  This means the full wavelength is \( \lambda = 75.0 \space \text{cm} \text{ or } 0.75 \space \text{m} \).

This understanding helps calculate wave speed and other properties important in analyzing sound behavior in physical scenarios.

The adiabatic constant, denoted as \( \gamma \), is vital in characterizing how gases respond to compression and expansion without heat exchange. For sound in air, knowing \( \gamma \) allows us to better understand how sound velocity varies with temperature and pressure.
In the formula for the speed of sound:\[ v = \sqrt{\frac{\gamma RT}{M}} \]

• \( R \) is the universal gas constant: \( 8.314 \space \text{J/mol K} \)
• \( T \) is the absolute temperature in Kelvin (77.0\( ^\circ \)C = 350 K)
• \( M \) is the molar mass of air \( 0.029 \space \text{kg/mol} \)

Given that the calculated speed of sound is 375 m/s, we rearrange the formula to solve for \( \gamma \):\[ \gamma = \frac{v^2 M}{RT} \]Replacing values results in:\[ \gamma = \frac{(375 \space \text{m/s})^2 \times 0.029 \space \text{kg/mol}}{8.314 \space \text{J/mol K} \times 350 \space \text{K}} \approx 1.40 \]This approximation is commonly associated with how air behaves under sound conduction, also reflecting typical values used in related physics calculations.