In a pipe opened at both ends, $n_{1}$ and $n_{2}$ be the frequencies corresponding to vibrating lengths $l_{1}$ and $l_{2}$ , respectively. The end correction is

\frac{n_{1} l_{1} - n_{2} l_{2}}{2 (n_{1} - n_{2})}

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\frac{n_{2} l_{2} - n_{1} l_{1}}{2 (n_{2} - n_{1})}

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\frac{n_{2} l_{2} - n_{1} l_{1}}{2 (n_{1} - n_{2})}

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\frac{n_{2} l_{2} - n_{1} l_{1}}{(n_{2} - n_{1})}

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Solution

The correct option is C $\frac{n_{2} l_{2} - n_{1} l_{1}}{2 (n_{1} - n_{2})}$
The frequency of the $p t h$ mode of vibration in an open organ pipe,
$ν = \frac{p v}{2 (L + 2 e)}$
Where, $e$ is the end correction of one side and $L$ is the length of the pipe

Given : For $L = l_{1}$ , frequency of $p t h$ mode $= n_{1}$
For $L = l_{1}$ , frequency of $p t h$ mode $= n_{1}$

$\Rightarrow$ $n_{1} = \frac{p v}{2 (l_{1} + 2 e)}$ .........(1)

$\Rightarrow n_{2} = \frac{p v}{2 (l_{2} + 2 e)}$ ..........(2)

Dividing (1) and (2), we get

$\frac{n_{1}}{n_{2}} = \frac{p v}{2 (l_{1} + 2 e)} \times \frac{2 (l_{2} + 2 e)}{p v}$

$∴$ $\frac{n_{1}}{n_{2}} = \frac{l_{2} + 2 e}{l_{1} + 2 e}$

$⟹$ $e = \frac{n_{2} l_{2} - n_{1} l_{1}}{2 (n_{1} - n_{2})}$

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If two lines have direction cosines

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n_{1} n_{2} = 2 n_{1} - n_{2}

n_{1}, n_{2} ϵ

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