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A hollow pipe of length 0.8 m is closed at one end. A uniform string of length 0.5 m, vibrating in its second harmonic at its open end, resonates with the fundamental frequency of the pipe. If the tension in the string is 50 N and the speed of sound in air is 320 m/s, find the mass of the string.

1. 5 g
2. 10 g
3. 20 g
4. 40 g

Correct answer: 10 g

Solution

The closed pipe's fundamental is 320/(4*0.8) = 100 Hz. The string's second harmonic is v_string/0.5 = 100 Hz, giving v_string = 50 m/s. Linear density mu = T/v_string² = 50/2500 = 0.02 kg/m, so mass = 0.02*0.5 = 0.01 kg = 10 g.

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