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 frequency of the second mode 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 need to first find the speed of the wave on the string, then find the frequency of the fundamental mode of the string, and finally use that to find the length of the organ pipe.

1. 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. Plugging in the given values, we get:

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

2. Find the frequency of the fundamental mode of the string: The frequency of the fundamental mode of a string fixed at both ends is given by f = v/2L, where L is the length of the string. Plugging in the values we have:

f = 5477.23 m/s / (2 * 0.4 m) = 6846.54 Hz

3. Find the length of the organ pipe: The frequency of the second mode of an organ pipe open at both ends is given by f = v/2L, where v is the speed of sound in air and L is the length of the pipe. We can rearrange this to solve for L:

L = v/2f = 331 m/s / (2 * 6846.54 Hz) = 0.0242 m

So the length of the organ pipe is 0.0242 m, or 2.42 cm.