Question

An organ pipe \(P_{1}\) closed at one end and vibrating in its first overtone and other pipe \(P_{2}\) open at both ends vibrating in its third overtone are in resonance with a given tuning fork. The ratio of length of \(P_{1}\) and \(P_{2}\) will be

• A. \(3:8\)
• B. \(3:4\)
• C. \(3:2\)
• D. \(3:1\)

Hint:

Write the frequency for the first overtone of a closed pipe and the third overtone of an open pipe and equate them.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Closed pipe: \(f_n=(2n+1)\dfrac{v}{4L}\). Open pipe: \(f_n=(n+1)\dfrac{v}{2L}\).

Step 2: Write the frequencies
Closed pipe, first overtone (3rd harmonic): \(f_1=\dfrac{3v}{4L_1}\).
Open pipe, third overtone (4th harmonic): \(f_2=\dfrac{4v}{2L_2}=\dfrac{2v}{L_2}\).

Step 3: Equate
\(\dfrac{3}{4L_1}=\dfrac{2}{L_2}\), so \(\dfrac{L_1}{L_2}=\dfrac{3}{8}\).

Step 4: Compare the options
The ratio is \(3:8\), option (A).

Final Answer:
The ratio of the lengths P1 to P2 is 3 to 8. \[ \boxed{\text{(A) }3:8} \]