Question

A pipe in an organ loft in Vancouver, British Columbia, is 0.700 m long, but its type (open‑open or open‑closed) is not specified. The speed of sound in air depends on the centigrade temperature T according to the following:

(T)=√109,700+402T

Find the temperature in the pipe if the ninth oo associated with the pipe that is resonating corresponds to 2.50×103 Hz.

Answer 1

To find the temperature in the pipe resonating at the ninth harmonic, we can use the formula for the speed of sound in air and the formula for the frequency of a pipe closed at one end.

Step-by-step explanation:

To find the temperature in the pipe, we can use the formula for the speed of sound in air given the temperature:

speed of sound = sqrt(109700 + 402T)

Since the pipe is resonating at the ninth harmonic (9th overtone) and the frequency is given as 2.50×10^3 Hz, we can use the formula for the frequency of a pipe closed at one end:

frequency = (2n-1) * (speed of sound) / (4L)

Substituting the given values, we can solve for T:

2.50×10^3 = (2*9-1) * sqrt(109700 + 402T) / (4 * 0.7)

Simplifying the equation and solving for T, we find that the temperature in the pipe is T = 43.62°C.

Answer 2

The length of an organ pipe closed at one end can be calculated using the formula Length = (n * wavelength) / 4. For the given fundamental frequency, the length of the pipe needed is 0.251 m.

Step-by-step explanation:

The length of an organ pipe closed at one end can be found using the formula:

Length = (n * wavelength) / 4

Where n is the harmonic number and wavelength is the distance between two consecutive nodes or antinodes in the pipe. For the fundamental frequency (n=1), the length of the pipe can be calculated as:

Length = (1 * wavelength) / 4

Substituting the given fundamental frequency of 262 Hz and the speed of sound in the air of 331 m/s, we can solve for the wavelength:

Wavelength = speed of sound/frequency

Once we have the wavelength, we can solve for the length:

Length = (1 * wavelength) / 4

Therefore, the length of the pipe needed to produce a fundamental frequency of 262 Hz when the air temperature is 20.0°C is:

Length = (1 * 331 m/s / 262 Hz) / 4 = 0.251 m