A string with a mass per length of 2.00 g/m is stretched with a force of 120 N between two points that are 0.400 m apart. The fundamental frequency of the stretched string is in tune with the fundamental frequency of an organ pipe filled with air and open at both ends. The speed of sound in air at 0°C is 331 m/s. What is the length of the organ pipe?

To solve this problem, we first need to find the speed of the wave on the string. The speed of a wave on a string is given by the formula v = sqrt(F/μ), where F is the tension in the string and μ is the mass per unit length.

Given: F = 120 N μ = 2.00 g/m = 0.002 kg/m

Substituting these values into the formula, we get:

v = sqrt(120 N / 0.002 kg/m) = 5477.23 m/s

Next, we need to find the wavelength of the wave on the string. The wavelength is twice the length of the string, because the string is fixed at both ends and thus forms a standing wave with a wavelength twice its length.

Given: Length of string = 0.400 m

So, wavelength = 2 * 0.400 m = 0.800 m

The frequency of the wave on the string is given by the formula f = v/λ, where v is the speed of the wave and λ is the wavelength.

Substituting the values we found earlier, we get:

f = 5477.23 m/s / 0.800 m = 6846.54 Hz

This is the frequency of the wave on the string, and it is also the frequency of the wave in the organ pipe, because we are told that the two are in tune with each other.

The speed of sound in air is given as 331 m/s. The wavelength of the sound in the organ pipe is given by the formula λ = v/f, where v is the speed of sound in air and f is the frequency.

Substituting the given and calculated values, we get:

λ = 331 m/s / 6846.54 Hz = 0.0483 m

Finally, the length of the organ pipe is half the wavelength, because the pipe is open at both ends and thus forms a standing wave with a wavelength twice its length.

So, length of organ pipe = 0.0483 m / 2 = 0.0241 m or 24.1 cm.