Two organ pipes having the same internal diameter ( d) but of different lengths `l_(1) " and " l_(2)` are closed at one end. If they are vibrating at their fundamental frequencies, then the end correction at end is given by

Options

A. `(n_(1)l_(1) - n_(2)l_(2))/(n_(2) - n_(1)) `
B. `(n_(2)l_(2) - n_(1) l_(1))/(n_(2) - n_(2)) `
C. `(n_(2) - n_(1))/(n_(1)l_(1) - n_(2)l_(2))`
D. `(n_(1)l_(2) - n_(2) l_(1))/(n_(1) - n_(2))`

Answer

A

Explanation

The effective length of the air columns in the two pipes are given by `L_(1) = l_(1) + e " and " L_(2) = l_(2) + e`
where e = end correction at the open end ` = 0.3d`
Since d is the same for both , e has the same value .
Let v be speed of sound in air , then the fundamental frequencies of the two pipes are
`n_(1) = v/(4L_(1)) = v/(4(l_(1) + e))`
` :. v = 4n_(1)(l_(1) + e)`....(1)
and ` n_(2) = v/(4L_(2)) = v/(4(l_(2) + e))`
` :. v = 4n_(2) (l_(2) + e) ` ...(2)
`:. ` from (1) and (2),
` v = 4n_(1) (l_(1) + e) = 4n_(2) (l_(2) + e) `
`:. n_(1) l_(1) + n_(1) e = n_(2) l_(2) + n_(2) e`
` :. n_(1) l_(1) - n_(2)l_(2)= (n_(2) - n_(1)) e`
` :. e = (n_(1)l_(1) - n_(2)l_(2))/(n_(2) - n_(1))`