Question

An organ pipe \(P_{1}\) closed at one end has vibrating air column in its first overtone and another pipe \(P_{2}\) open at both ends has vibrating air column in the third overtone, are in resonance with a given tuning fork. The ratio of the length of pipe \(P_{1}\) to that of \(P_{2}\) is

• A. \(1:8\)
• B. \(3:8\)
• C. \(5:8\)
• D. \(1:2\)

Hint:

Write the resonant frequency of each pipe for the stated overtone and equate them.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
A closed pipe in its first overtone vibrates as the 3rd harmonic: \(f=\dfrac{3v}{4L_{1}}\). An open pipe in its third overtone vibrates as the 4th harmonic: \(f=\dfrac{4v}{2L_{2}}=\dfrac{2v}{L_{2}}\).

Step 2: Equate the frequencies
Both resonate with the same fork, so \(\dfrac{3v}{4L_{1}}=\dfrac{2v}{L_{2}}\).

Step 3: Calculation
\(\dfrac{L_{1}}{L_{2}}=\dfrac{3}{8}\)

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