Question

A pipe open at both ends has a frequency \(f\) in air. The pipe is dipped vertically in water, so that its \((1/3)\) part of its length is in water. The fundamental frequency of the air column is now \(xf\), the value of \(x\) is

• A. \(\frac{1}{2}\)
• B. \(\frac{3}{4}\)
• C. \(\frac{3}{2}\)
• D. \(1\)

Hint:

Once one third is under water the pipe behaves as a closed pipe of length two thirds of the original.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
An open pipe of length \(L\) has fundamental frequency \(f=\frac{v}{2L}\).

Step 2: After dipping
When one third of the length is in water, the air column has length \(\frac{2L}{3}\) and is closed at one end.

Step 3: Calculation
The fundamental frequency of a closed pipe is \(f'=\frac{v}{4\left(\frac{2L}{3}\right)}=\frac{3v}{8L}\).

Step 4: Compare with f
\(\frac{f'}{f}=\frac{3v/8L}{v/2L}=\frac{3}{4}\), so \(x=\frac{3}{4}\), option (B).

Final Answer:
The value of \(x\) is \(\frac{3}{4}\). \[ \boxed{\text{(B) }\dfrac{3}{4}} \]