Question

A pipe 1.00 m long is closed at one end. A guitar is placed near the open end of the tube and the string is plucked. The guitar string is 0.50 meters long and has a mass of 0.0010 kg. The string vibrates in its fundamental mode (1st harmonic) and produces a 1st overtone (3rd harmonic) standing wave in the closed pipe. If the speed of sound is 340 m/s, determine the tension in the guitar string.

Hint: the frequency of the vibrating guitar string is the same as the frequency produced in the closed pipe.

Answer 1

To determine the tension in the guitar string, we can use the formula for the fundamental frequency of the closed pipe and the speed of sound. By substituting the known values into the formula, we can solve for the tension in the guitar string.

Step-by-step explanation:

To determine the tension in the guitar string, we need to find the frequency of the standing wave in the closed pipe. Since the open end of the pipe is near the guitar string, it will produce the same frequency as the string.

The frequency of a standing wave can be determined using the formula:

• f = (v / λ) * n

where f is the frequency, v is the speed of sound, λ is the wavelength, and n is the harmonic number. In this case, the closed pipe is in the 3rd harmonic, so n = 3. The fundamental frequency (1st harmonic) of the closed pipe is 1/4 of the 3rd harmonic.

The formula for the fundamental frequency is:

• f1 = v / 4L

where f1 is the fundamental frequency, v is the speed of sound, and L is the length of the closed pipe.

Substituting the known values into the formula, we can solve for the tension in the guitar string.