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

Question

A pipe open at both ends has a fundamental frequency $f$ in air. The pipe is now dipped vertically in water drum to half of its length. The fundamental frequency of the air column is now equal to:

Answer 1

Step 1: Wavelength relations
Closed pipe, first overtone: $L_1=\dfrac{3\lambda}{4}$.
Open pipe, third overtone: $L_2=\dfrac{4\lambda}{2}=2\lambda$.

Step 2: Same fork
Both pipes resonate with the same fork, so $\lambda$ is the same.

Step 3: Ratio
$\dfrac{L_1}{L_2}=\dfrac{3/4}{2}=\dfrac{3}{8}$, option (A).

Final Answer:
The lengths are in the ratio 3 to 8. \[ \boxed{\text{(A) }3:8} \]

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

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The Correct Option is A

Solution and Explanation

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