Question

The second overtone of a closed organ pipe $P_1$ and the third overtone of an open pipe $P_2$ are in unison with a tuning fork. The ratio of the length of $P_1$ and $P_2$ will be

• A. $5:8$
• B. $1:3$
• C. $8:3$
• D. $3:1$

Answer 1

The correct option is A $5:8$

Let $v$ be the velocity of sound.

Let $L_1$ and $L_2$ be the lengths of the closed and open pipe respectively.

Frequency of mth harmonic of an open pipe $P_2$ ${\nu }_m=\frac{mv}{2L_2}$

Frequency of 3rd overtone $(m=4)$ ${\nu }_4=\frac{4v}{2L_2}$

Frequency of different modes of vibration in a closed pipe $P_1$

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

Frequency of 2nd overtone $(n=3)$ ${\nu }_3^{\prime }=\frac{5v}{4L_1}$

Given ${\nu }_3^{\prime }={\nu }_4$

$⟹L_1:L_2=5:8$