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 ${\displaystyle \frac{m}{s}}$, then the mass of the string is near ______ 

पर्याय

• 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 ${\displaystyle \frac{m}{s}}$, then the mass of the string is near 125 grams.

Explanation:

Fundamental frequency of the pipe

${\displaystyle n=\frac{V}{4L}=\frac{340}{4\times 1}}$ = 85 Hz

∴ Third hannonic of the string = 85 Hz

∴ ${\displaystyle 85=\frac{3}{2\ell }\sqrt{\frac{T}{m}}=\frac{3}{2\times 0.25}\sqrt{\frac{100}{m}}}$

∴ ${\displaystyle 85=\frac{3}{0.5}\times \frac{10}{\sqrt{m}}}$

∴ ${\displaystyle \sqrt{m}=\frac{3\times 10}{85\times 0.5}\approx 0.7}$

∴ m = 0.49 kg ≈ 0.5 kg/m

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