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