Question

A closed and an open organ pipe have the same length. When they are vibrating simultaneously in their first overtone, they produce three beats. The length of the open pipe is now made one third the original length and one of its ends is closed. On the other hand, the length of the closed pipe is made three times the original length. The number of beats produced when they vibrate with fundamental frequencies will be

• A. 8
• B. 14
• C. 17
• D. 20

Answer 1

Answer: (A)

$V=$ speed of sound and $L=$ lenght of pipes
For open pipe first overtone $f_{1}=\frac{V}{L}$
For closed pipe first overtone $f'_{1}=\frac{3 V}{4 L}$
$\therefore \, f_{1}-f'_{1}=\frac{V}{L}-\frac{3 V}{4 L}=3$
$\therefore \, \, \, \frac{V}{4 L}=3$
$\therefore \, \, \, \frac{V}{L}=12 \, $
When we close one end of the open pipe and make its length one-third of the original length, then its fundamental frequency is
$f=\frac{V}{4 \left(\right. \frac{L}{3} \left.\right)}=\frac{3 V}{4 L}$
When the length of closed pipe is made $3$ times, the fundamental frequency is
$f'=\frac{V}{4 \left(\right. 3 L \left.\right)}=\frac{V}{12 L}$
Beats produced $=f-f'$
$=\frac{3 V}{4 L}-\frac{V}{12 L}$
$=\frac{8}{12}.\frac{V}{L}=\frac{8}{12}\times 12$
$=8$