Question

# An organ pipe $$P_1$$ closed at one end vibrating in its first harmonic and pipe $$P_2$$ open at both ends vibrating in its third harmonic are in resonance with the same tuning fork. The ratio of their lengths is

A
3/8
B
3/4
C
1/8
D
1/6

Solution

## The correct option is D $$1/6$$Let $$v$$ be the velocity of sound.Closed organ pipe $$P_1$$ of length $$L_1$$:Frequency of different modes of vibration    $$\nu'_n = \dfrac{(2n - 1) v}{4 L_1}$$First harmonic i.e $$n =1$$,              $$\nu'_1 = \dfrac{v}{4 L_1}$$Open organ pipe $$P_2$$ of length $$L_2$$:Frequency of mth harmonic           $$\nu_m = \dfrac{m v}{4 L_2}$$For third harmonic i.e  $$m = 3$$                     $$\nu_3 = \dfrac{3 v}{2 L_2}$$But       $$\nu'_1 = \nu_3$$$$\dfrac{v}{ 4 L_1} = \dfrac{3 v}{2 L_2 } \implies \dfrac{L_1}{L_2} = \dfrac{1}{6}$$Physics

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