The fundamental frequency in an open organ pipe is equal to the third harmonic of a closed organ pipe. If the length of the closed organ pipe is 20 cm, the length of the open organ pipe is [2018]

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Solution

Q.
The fundamental frequency in an open organ pipe is equal to the third harmonic of a closed organ pipe. If the length of the closed organ pipe is 20 cm, the length of the open organ pipe is [2018]

1 13.2 cm

2 8 cm

3 12.5 cm

4 16 cm

Ans.
(1)

For closed organ pipe, third harmonic is $\frac{3 v}{4 l}$

For open organ pipe, fundamental frequency is $\frac{v}{2 l'}$

Given, third harmonic for closed organ pipe = fundamental frequency for open organ pipe.

$∴$ $\frac{3 v}{4 l} = \frac{v}{2 l'} \Rightarrow l' = \frac{4 l}{3 \times 2} = \frac{2 l}{3};$

where $l$ and $l'$ are the lengths of closed and open organ pipes respectively.

$∴$ $l' = \frac{2 \times 20}{3} = 13.33$ $cm$

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