MHT CET · Physics · Waves and Sound

The fundamental frequency of an air column in pipe ' \(\mathrm{A}\) ' closed at one end coincides with second overtone of pipe ' \(\mathrm{B}\) ' open at both ends. The ratio of length of pipe ' \(\mathrm{A}\) ' to that of pipe ' \(\mathrm{B}\) ' is

A \(3: 8\)
B \(3: 4\)
C \(1: 6\)
D \(2: 3\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(1: 6\)

Step-by-step Solution

Detailed explanation

Fundamental frequency of pipe closed at one end
\(
\mathrm{n}=\frac{\mathrm{v}}{4 \ell}
\)
Second overtone of pipe open at both the ends
\(
\begin{aligned}
& \mathrm{n}^{\prime}=\left(\frac{3 \mathrm{v}}{2 \ell^{\prime}}\right) \\
& \mathrm{n}^{\prime}=\mathrm{n} \\
& \therefore \frac{3 \mathrm{v}}{2 \ell^{\prime}}=\frac{\mathrm{v}}{4 \ell} \\
& \therefore \frac{\ell}{\ell^{\prime}}=\frac{1}{6}
\end{aligned}
\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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