Waves - Result Question 49

51. An organ pipe $P_1$ closed at one end vibrating in its first overtone and another pipe $P_2$, open at both ends vibrating in its third overtone are in resonance with a given tuning fork. The ratio of lengths of $P_1$ and $P_2$ respectively are given by

[1997]

(a) $1: 2$

(b) $1: 3$

(c) $3: 8$

(d) $3: 4$

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Answer:

Correct Answer: 51. (c)

Solution:

  1. (c) We know that the length of pipe closed at one end for first overtone $(l_1)=\frac{3 \lambda}{4}$

and length of the open pipe for third overtone $(l_2)=\frac{4 \lambda}{2}=2 \lambda$.

Therefore, the ratio of lengths $\frac{l_1}{l_2}=\frac{3 \lambda / 4}{2 \lambda}=\frac{3}{8}$ or $l_1: l_2=3: 8$.

In an open pipe all harmonics are present whereas in a closed organ pipe only alternate harmonies of frequencies are present. Hence musical sound produced by an open organ pipe is sweeter than that produced by a closed organ pipe.