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Question
An organ pipe of length L open at both ends is found to vibrate in its first harmonic when sounded with a tuning fork of 480 Hz. What should be the length of a pipe closed at one end, so that it also vibrates in its first harmonic with the same tuning fork?
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Solution
For the first harmonic of open organ pipe L = `λ/2`
⇒ λ = 2L
⇒ `v/f_0` = 2L
⇒ `(f_0)_"open" = v/(2L)`
Where v is the speed of the sound wave in air.
For the first harmonic of closed organ pipe L' = `λ/4`
⇒ λ = 4L'
⇒ `v/f_0` = 4L'
⇒ `(f_0)_"close" = v/(4L^')`
⇒ `v/(2L) = v/(4L^')` .......[∵ Speed remains constant]
⇒ `L^'/L = 2/4 = 1/4`
⇒ `L^' = L/2`
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