Question

\(n_{1}\) is the fundamental frequency of the pipe closed at one end and \(n_{2}\) is the fundamental frequency of the pipe open at both ends. If both are joined end to end, the fundamental frequency of closed pipe so formed is

• A. \(\frac{n_{1}n_{2}}{n_{2}+2n_{1}}\)
• B. \(\frac{n_{1}n_{2}}{2n_{2}+n_{1}}\)
• C. \(\frac{n_{1}+2n_{2}}{n_{1}n_{2}}\)
• D. \(\frac{2n_{1}+n_{2}}{n_{1}n_{2}}\)

Hint:

Write each fundamental in terms of its length; the joined pipe is closed with length equal to the sum.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Closed pipe of length \(L_{1}\): \(n_{1}=\dfrac{v}{4L_{1}}\). Open pipe of length \(L_{2}\): \(n_{2}=\dfrac{v}{2L_{2}}\).

Step 2: Lengths
\(L_{1}=\dfrac{v}{4n_{1}}\) and \(L_{2}=\dfrac{v}{2n_{2}}\).

Step 3: Joined Pipe
The combined pipe is closed at one end with length \(L_{1}+L_{2}\):
\(n=\dfrac{v}{4(L_{1}+L_{2})}=\dfrac{v}{\dfrac{v}{n_{1}}+\dfrac{2v}{n_{2}}}=\dfrac{1}{\dfrac{1}{n_{1}}+\dfrac{2}{n_{2}}}\)

Step 4: Simplify
\(n=\dfrac{n_{1}n_{2}}{n_{2}+2n_{1}}\), option (A).

Final Answer:
The fundamental frequency of the joined closed pipe is \(\dfrac{n_{1}n_{2}}{n_{2}+2n_{1}}\). \[ \boxed{\text{(A) }\dfrac{n_{1}n_{2}}{n_{2}+2n_{1}}} \]