Question

A closed pipe and an open pipe of same length produce $2$ beats, when they are set into vibrations simultaneously in their fundamental mode. If the length of the open pipe is halved, and that of closed pipe is doubled, and if they are vibrating in the fundamental mode, then the number of beats produced is:

• A. $4$
• B. $7$
• C. $2$
• D. $8$

Answer 1

The correct option is B $7$

Let the speed of sound in air be $v$ .

Beat frequency when both pipes have same length $b={\nu }_1-{\nu }_1^{\prime }=2$

Fundamental frequency in an open pipe ${\nu }_1=\frac{v}{2L}$

Fundamental frequency in a closed pipe ${\nu }_1^{\prime }=\frac{v}{4L}$

$\therefore$ $\frac{v}{2L}$ $-\frac{v}{4L}=2$

$⟹$ $\frac{v}{L}=8$ ..........(1)

Now the length of the open pipe is halved and that of closed pipe is doubled.

New beat frequency $b^{\prime }=\frac{v}{2\left(L/2\right)}-\frac{v}{4\left(2L\right)}=\frac{7}{8}\frac{v}{L}$

$⟹$ $b^{\prime }=\frac{7}{8}\times 8=7$