MHT CET · Physics · Waves and Sound
\(n_{1}\) is the frequency of the pipe closed at one end and \(n_{2}\) is the frequency of the pipe open at both ends. If both are joined end to end, find the fundamental frequency of closed pipe so formed
- A \(\frac{n_{1} n_{2}}{n_{2}+2 n_{1}}\)
- B \(\frac{n_{1} n_{2}}{2 n_{2}+n_{1}}\)
- C \(\frac{n_{1}+2 n_{2}}{n_{2} n_{1}}\)
- D \(\frac{2 n_{1}+n_{2}}{n_{2} n_{1}}\)
Answer & Solution
Correct Answer
(A) \(\frac{n_{1} n_{2}}{n_{2}+2 n_{1}}\)
Step-by-step Solution
Detailed explanation
Frequency of closed pipe, \(n_{1}=\frac{v}{4 l_{1}} \Rightarrow l_{1}=\frac{v}{4 n_{1}}\)
Frequency of open pipe, \(n_{2}=\frac{v}{2 l_{1}} \Rightarrow l_{2}=\frac{v}{2 n_{2}}\)
when both pipe are joined then length of closed pipe
\(
\begin{array}{c}
l=l_{1}+l_{2} \\
\frac{v}{4 n}=\frac{v}{4 n_{1}}+\frac{v}{2 n_{2}}
\end{array}
\)
or \(\quad \frac{1}{2 n}=\frac{1}{2 n_{1}}+\frac{1}{n_{2}}\)
or
\(
\frac{1}{2 n}=\frac{n_{2}+2 n_{1}}{2 n_{1} n_{2}}
\)
or
\(
n=\frac{n_{1} n_{2}}{n_{2}+2 n_{1}}
\)
Frequency of open pipe, \(n_{2}=\frac{v}{2 l_{1}} \Rightarrow l_{2}=\frac{v}{2 n_{2}}\)
when both pipe are joined then length of closed pipe
\(
\begin{array}{c}
l=l_{1}+l_{2} \\
\frac{v}{4 n}=\frac{v}{4 n_{1}}+\frac{v}{2 n_{2}}
\end{array}
\)
or \(\quad \frac{1}{2 n}=\frac{1}{2 n_{1}}+\frac{1}{n_{2}}\)
or
\(
\frac{1}{2 n}=\frac{n_{2}+2 n_{1}}{2 n_{1} n_{2}}
\)
or
\(
n=\frac{n_{1} n_{2}}{n_{2}+2 n_{1}}
\)
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