Q: An organ pipe P1 closed at one end vibrating in its first overtone and another pipe P2 open at both ends vibrating in third overtone are in resonance with a given tuning fork. The ratio of the length of P1 to that of P2 is
(a) 8/3
(b) 3/8
(c) 1/2
(d) 1/3
Ans: (b)
Sol: Frequency of first overtone for closed organ pipe P1 is
$ \displaystyle \nu_1 = (2\times 1 + 1)\frac{v}{4l_1} $
$ \displaystyle \nu_1 = \frac{3 v}{4l_1} $
Frequency of third overtone for open organ pipe P2 is
$ \displaystyle \nu_2 = \frac{4 v}{2l_2} $
According to question ,
ν1 = ν2
$ \displaystyle \frac{3 v}{4l_1} = \frac{4 v}{2l_2} $
16 l1 = 6 l2
$ \displaystyle \frac{l_1}{l_2} = \frac{3}{8} $