Question

Closed and open organ pipes have same length. In first overtone, they produce four beats. If open pipe length is halved and closed pipe doubled, the beats in fundamental mode are

• A. 8
• B. 10
• C. 14
• D. 16

Hint:

Fundamental of closed pipe $= v/4L$; Fundamental of open pipe $= v/2L$.

Answer 1

The Correct Option is B

Step 1: Initial Frequencies
- Closed pipe 1st overtone $f_{c1} = 3v/4L$. - Open pipe 1st overtone $f_{o1} = 2v/2L = v/L$. - Beats: $|v/L - 3v/4L| = 4 \implies v/4L = 4 \implies v/L = 16$.

Step 2: New Fundamentals
- New Open pipe ($L' = L/2$): $f'_o = v/2(L/2) = v/L = 16$. - New Closed pipe ($L'' = 2L$): $f'_c = v/4(2L) = v/8L = 16/8 = 2$.

Step 3: Calculation
New beats: $|16 - 2| = 14$.

Step 4: Conclusion
Hence, the number of beats is 14. Final Answer: (C)