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Question : 17 of 27
Marks: +1, -0
A pipe 20 cm long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a 430 Hz source? Will the same source be in resonance with the pipe if both ends are open? (speed of sound in air is 340 m s−1340\,\text{m}\,\text{s}^{-1}).
Solution:  
Here, L = 20 cm = 0.2 m, υn=430\upsilon_n = 430 Hz, v=340 m s−1v = 340\,\text{m}\,\text{s}^{-1}
The frequency of nth normal mode of vibration of closed pipe is
υn=(2n−1)v4L\upsilon_n = (2n-1)\frac{v}{4L}
∴430=(2n−1)3404×0.2\therefore 430 = (2n-1)\frac{340}{4 \times 0.2} or 2n−1=430×4×0.2340=1.022n-1 = \frac{430 \times 4 \times 0.2}{340} = 1.02
2n=2.022n = 2.02 or n=1.01n = 1.01
Hence it will be the lst normal mode of vibration.
In a pipe, open at both ends, we have
υn=n×v2L⇒n×3402×0.2=430\upsilon_n = n \times \frac{v}{2L} \Rightarrow \frac{n \times 340}{2 \times 0.2} = 430
∴n=430×2×0.2340=0.5\therefore n = \frac{430 \times 2 \times 0.2}{340} = 0.5
As n has to be an integer, therefore, open organ pipe cannot be in resonance with the source.
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