Question

The closed and open organ pipes have same length. When they are vibrating simultaneously in first overtone, they produce four beats. The length of open pipe is made half and that of the closed pipe is made two times the original. Now the number of beats produced if the two pipes are vibrating in their fundamental modes simultaneously is

• A. 8
• B. 10
• C. 14
• D. 16

Hint:

Use the first overtone frequencies to find v/L, then recompute the fundamental frequencies for the new lengths.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Closed pipe: fundamental \(\dfrac{v}{4L}\), first overtone \(\dfrac{3v}{4L}\). Open pipe: fundamental \(\dfrac{v}{2L}\), first overtone \(\dfrac{v}{L}\).

Step 2: Use the first beat data
Beats \(=\dfrac{v}{L}-\dfrac{3v}{4L}=\dfrac{v}{4L}=4\), so \(\dfrac{v}{L}=16\).

Step 3: New lengths
Open pipe of length \(\dfrac{L}{2}\): \(f=\dfrac{v}{2(L/2)}=\dfrac{v}{L}=16\text{ Hz}\).
Closed pipe of length \(2L\): \(f=\dfrac{v}{4(2L)}=\dfrac{v}{8L}=2\text{ Hz}\).

Step 4: Calculation
Number of beats \(=16-2=14\).

Final Answer:
The number of beats produced is 14. \[ \boxed{\text{(C) }14} \]