Question

A pipe closed at one end has a length of 1m and at its open end, 0.25 m long uniform string is vibrating in its third harmonic and resonates with a fundamental frequency of pipe. If the tension in the string is 100 N and the speed of sound is 340 `m/s`, then the mass of the string is near ______ 

Options

• 125 gram

• 200 gram

• 150 gram

• 175 gram

Answer 1

A pipe closed at one end has a length of 1m and at its open end, 0.25 m long uniform string is vibrating in its third harmonic and resonates with a fundamental frequency of pipe. If the tension in the string is 100 N and the speed of sound is 340 `m/s`, then the mass of the string is near 125 grams.

Explanation:

Fundamental frequency of the pipe

`n = V/(4L) = 340/(4 xx 1)` = 85 Hz

∴ Third hannonic of the string = 85 Hz

∴ `85 = 3/(2ℓ)sqrt(T/m) = 3/(2 xx 0.25) sqrt(100/m)`

∴ `85 = 3/0.5 xx 10/sqrtm`

∴ `sqrtm = (3 xx 10)/(85 xx 0.5) ≈ 0.7`

∴ m = 0.49 kg ≈ 0.5 kg/m

Mass of the wire M = mL = 0.5 × 0.25 = 0.125 kg = 125 g