A closed organ pipe of length `L` and an open organ pipe contain gass of densities `rho_(1)` and `rho_(2)`, respectively. The compressibility of gass are equal in both the pipes. Both the pipes are vibrating in their first overtone with same frequency . The length of the open orange pipe is
(a) `(L)/(3)`
`(4l) / (3)`
(c ) `(4l)/(3)sqrt((rho_(1))/(rho_(2)))`
(d) `(4l)/(3)sqrt((rho_(2))/(rho_(1)))`

Options

A. `(L)/(3)`
B. `(4L)/(3)`
C. `(4L)/(3)sqrt((rho_(1))/(rho_(2)))`
D. `(4L)/(3)sqrt((rho_(2))/(rho_(1)))`

Answer

C

Explanation

Frequency of first over tone ofclosed organ pipe = Frequency of first over tone ofo pen organ pipe
`(3v)/(4L_(1)) = (v)/(L_(2))`
`(3)/(4L_(1))sqrt((gammaP)/(rho_(1))) = (1)/(L_(2))sqrt((gammaP)/(rho_(2)) ) [As v = sqrt((gammaP)/(rho))]`
implies `L_(2) = (4L_(1))/(3)sqrt((rho_(1))/(rho_(2))) = (4L)/(3)sqrt((rho_(1))/(rho_(2)))`