Question

A certain organ pipe, open at both ends, produces a fundamental frequency of 300 Hz in air. If the pipe is filled with helium at the same temperature, what fundamental frequency fHe will it produce? Take the molar mass of air to be 28.8 g/mol and the molar mass of helium to be 4.00 g/mol. Now consider a pipe that is stopped (i.e., closed at one end) but still has a fundamental frequency of 300 Hz in air. How does your answer to first question, fHe, change?

Answer 1

The student is inquiring about how changing the gas inside an organ pipe to helium would affect its fundamental frequency. By calculating the ratio of the speeds of sound in helium and air using the molar masses and applying it to the frequency produced in air, we determine that the frequency would be higher in helium for both open and closed-end pipes.

Step-by-step explanation:

The student is asking about the effect of changing the gas inside an organ pipe on its fundamental frequency. The fundamental frequency in an organ pipe open at both ends is determined by the length of the pipe and the speed of sound within the gas that fills the pipe. The speed of sound in a gas is dependent on the properties of that gas, specifically its temperature and its molar mass. The frequency of sound produced by a pipe filled with a different gas can be found using the relation f = v / (2L), where f is the frequency, L is the length of the pipe, and v is the speed of sound in the gas.

As the speed of sound v in a gas is given by v = √(γRT/M), where γ is the adiabatic index of the gas, R is the ideal gas constant, T is the absolute temperature, and M is the molar mass of the gas, the frequency of sound in helium can be found. Since helium has a molar mass much smaller than air, the speed of sound in helium is greater and thus the pipe will produce a higher frequency in helium. The relationship between the frequencies can be determined by the ratio of the speeds of sound in helium and air, which can be simplified to fHe = fair √(Mair/MHe).

For a pipe closed at one end, the calculation differs as the fundamental frequency is f = v / (4L). However, the relationship between the frequencies with air and helium remains the same. Thus, when the pipe is closed at one end and filled with helium, it will still produce a fundamental frequency higher than that with air.

Answer 2

The fundamental frequency of an organ pipe is determined by its length and the speed of sound in the medium inside the pipe. The frequency in air can be calculated using the equation f = (v / 2L). If the pipe is filled with helium, the speed of sound will change, and the new fundamental frequency in helium can be calculated as fHe = (sqrt(γRT/MHe) / 2LHe). When the pipe is closed at one end, the fundamental frequency becomes the second harmonic, so fHe would become the new fundamental frequency.

Step-by-step explanation:

The fundamental frequency of an organ pipe is determined by its length and the speed of sound in the medium inside the pipe. When the pipe is open at both ends, the fundamental frequency is given by the equation:

f = (v / λ) = (v / 2L)

where f is the frequency, v is the speed of sound, and L is the length of the pipe. In this case, the fundamental frequency in air is 300 Hz. So, substituting the values, we get:

300 Hz = (v / 2L)

Now, if we fill the pipe with helium, the speed of sound will change due to the difference in molar mass between air and helium. The speed of sound in a gas is given by the equation:

v = sqrt(γRT/M)

where γ is the adiabatic index, R is the gas constant, T is the temperature, and M is the molar mass of the gas. Since we are assuming the temperature remains the same, the speed of sound in helium can be written as:

vHe = sqrt(γRT/MHe)

Now, we can calculate the new length needed to produce a fundamental frequency of fHe in helium:

fHe = (vHe / 2LHe)

Substituting the values, we get:

fHe = (sqrt(γRT/MHe) / 2LHe)

To find out how the frequency changes when the pipe is closed at one end, we need to consider the change in harmonics. When the pipe is open at both ends, the fundamental frequency is the first harmonic. But when the pipe is stopped, the fundamental frequency becomes the second harmonic. This means that the first overtone, or the second harmonic, is now the fundamental frequency. So, the answer to the second question, fHe becomes the new fundamental frequency when the pipe is closed at one end.