A long tube contains air at a pressure of 1.00 atm and a...

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 air 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?

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 air 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?

Key Concepts

End Correction

End correction refers to the phenomenon where the displacement antinode of a standing wave in an open tube occurs slightly outside the physical end of the tube. This adjustment is necessary because the air just outside the tube also vibrates, effectively extending the length of the resonant air column. Including the end correction leads to a more accurate determination of the wavelengths and resonant conditions in the tube, an essential detail when precise measurements are required in acoustic experiments.

Standing Waves in Open-Closed Tubes

In an open-closed tube, standing waves are established when the wave reflects such that a displacement node occurs at the closed end and a displacement antinode at the open end. The resonant frequencies of these systems are determined by the boundary conditions, which dictate that the permissible wavelengths are related to the effective length of the tube. This concept is central in analyzing resonance phenomena and deriving the relationship between tube length and the wavelengths of the harmonics generated within the tube.

Speed of Sound in Air

The speed of sound in a medium like air relates the frequency of a sound wave and its wavelength through the equation v = f?. By observing the resonant positions in the tube, one can determine the wavelength of the standing wave and, with the known frequency of the source (in this case, the tuning fork), calculate the speed of sound. This concept is important not only for characterizing the medium's properties but also for understanding wave propagation in different conditions, such as varying temperatures.

Adiabatic Index (?)

The adiabatic index, ?, is the ratio of specific heats at constant pressure and constant volume for a gas. It plays a critical role in determining the speed of sound in an ideal gas, which is given by the formula v = ?(?RT/M) where R is the universal gas constant, T is the absolute temperature, and M is the molar mass of the gas. By measuring the speed of sound under controlled conditions, one can back-calculate the value of ?, linking experimental observations to the thermodynamic properties of the gas.

Transcript

00:01 So in this problem, we have a tube that's closed by a piston here shown in red.

00:10 And at the end of the piston, there's an opening in the tube.

00:17 So the end opposite where the piston is, where there's a tuning fork right there that's producing a frequency of 500 hertz.

00:26 We know the pressure is one atmosphere, and we know the temperature is 77 degrees celsius.

00:32 So the first part asks us to determine the speed of sound, given that there is a loud noise heard at these distances, 8 centimeters, 55 .5 centimeters and 93 .0 centimeters.

00:50 18 centimeters, i should say.

00:54 So the realization that you have to make is that if this is the wave in the tube, when the piston moves, it's basically moving between nodes.

01:12 So the piston is solid, which means it's like a closed end, which means that for there to be a loud sound, the piston has to be at a node because the amplitude has to die at a node.

01:28 So that means that when the piston moves from here to here, that's a length of half the wavelength of the wave.

01:46 You see.

01:47 So in our case, the piston, let's say it moves from 18 .0 centimeters to 55 .5 centimeters.

02:04 That means that lambda over two, so that half of a wavelength, is 37 .5 centimeters.

02:17 Right? that's just the difference, or the difference and distance between those two measurements.

02:24 So this means that lambda is 0 .750 meters.

02:31 But then once we know the wavelength, we can find the speed quickly using v equals f lambda, the universal wave equation.

02:39 So the frequency is 0 .750 meters.

02:46 The wavelength, excuse me, that's the wavelength.

02:50 The frequency is 500 hertz and the wavelength is 0 .750 meters and that is 375 meters per second so there's the speed of sound in the tube what's gamma well to find gamma we can just use the formula so v is gamma r t over m so this is the speed gamma is the ratio of heat capacities.

03:26 That's what we're trying to find.

03:29 R is the ideal gas constant.

03:31 T is the temperature, which we're given.

03:34 And m is the molar mass of the medium...

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