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. $1/6$

Answer 1

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 ${\nu }_n^{\prime }=\frac{(2n-1)v}{4L_1}$

First harmonic i.e $n=1$, ${\nu }_1^{\prime }=\frac{v}{4L_1}$

Open organ pipe $P_2$ of length $L_2$:

Frequency of mth harmonic ${\nu }_m=\frac{mv}{4L_2}$

For third harmonic i.e $m=3$ ${\nu }_3=\frac{3v}{2L_2}$

But ${\nu }_1^{\prime }={\nu }_3$

$\frac{v}{4L_1}=\frac{3v}{2L_2}⟹\frac{L_1}{L_2}=\frac{1}{6}$