Q: An organ pipe P_{1} closed at one end vibrating in its first overtone and another pipe P_{2} open at both ends vibrating in third overtone are in resonance with a given tuning fork. The ratio of the length of P_{1} to that of P_{2} is

(a) 8/3

(b) 3/8

(c) 1/2

(d) 1/3

Ans: (b)

Sol: Frequency of first overtone for closed organ pipe P_{1} 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 P_{2} is

$ \displaystyle \nu_2 = \frac{4 v}{2l_2} $

According to question ,

ν_{1} = ν_{2}

$ \displaystyle \frac{3 v}{4l_1} = \frac{4 v}{2l_2} $

16 l_{1} = 6 l_{2}

$ \displaystyle \frac{l_1}{l_2} = \frac{3}{8} $