MHT CET · Physics · Waves and Sound

The fundamental frequency of an air column in a pipe open at both ends is ' \(\mathrm{f}_1\) '. Now \(80 \%\) of its length is immersed in water, the fundamental frequency of the air column becomes \(f_2\). The ratio of \(f_1: f_2\) is

A \(5: 2\)
B \(5: 4\)
C \(4: 5\)
D \(2: 5\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(2: 5\)

Step-by-step Solution

Detailed explanation

The fundamental frequency of the open pipe
\(=\mathrm{f}_1=\frac{\mathrm{V}}{2 \dot{\mathrm{~L}}}\)
After dipping in water, the pipe behaves like a closed pipe. As the pipe is dipped \(80 \%\) in water,
\(l=\frac{20}{100} \times \mathrm{L}=\frac{\mathrm{L}}{5}\)
\(\therefore \quad\) Fundamental frequency \(\mathrm{f}_2=\frac{\mathrm{V}}{4 l}=\frac{5 \mathrm{~V}}{4 \mathrm{~L}}\)
\(\therefore \quad \frac{\mathrm{f}_1}{\mathrm{f}_2}=\frac{\mathrm{V}}{2 \mathrm{~L}} \times \frac{4 \mathrm{~L}}{5 \mathrm{~V}}=\frac{2}{5}\)

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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