The closed and open organ pipes have same length. When they are vibrating simultaneously in first overtone, produce three beats. The length of open pipe is made `1/3` rd and closed pipe is made thre time the original , the number of beats produced will be

Options

A. 8
B. 14
C. 17
D. 20

Answer

C

Explanation

For open pipe first overtone ,
`v_(1) = v/L`
For closed pipe first overtone,
`v_(1) = (3v)/(4L)`
`:. v_(1) - v_(1) = (v)/(L) - (3v)/(4L) = 3`
`rArr (v)/(3L) = 3`
`:. v/L = 9 "…"(i)`
When length of open pipe is made `L/3`, then fundamental frequency
`v = (v)/(2(L/3)) = (3v)/(2L)`
When length of closed pipe is made 3 times, then fundamental frequency
`v' = (v)/(4(3L)) = v/(12L)`
Beat produced `= v = v'`
`= (3v)/(2L) - (v)/(12 L) = (17)/(12). (v)/(L)`
`= (17)/(12) xx 12` [ `:'` from Eqs. `(i)`]
`= 17`