MHT CET · Physics · Waves and Sound

A pipe open at both ends and a pipe closed at one end have save same length. The ratio of frequencies of air columns in their \(\mathrm{p}^{\text {th }}\) overtone respectively is

A \(\frac{p}{2 p+1}\)
B \(\frac{p+1}{2 p}\)
C \(\frac{p+1}{2 p+1}\)
D \(\frac{2(p+1)}{2 p+1}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{2(p+1)}{2 p+1}\)

Step-by-step Solution

Detailed explanation

Let \(l\) be the length of the pipe and \(v\) the speed of the sound. The frequency of the open organ pipe of \(\mathrm{p}^{\text {th }}\) overtone is,
\(\mathrm{f}_0=(\mathrm{p}+1) \frac{\mathrm{v}}{21}\)
And frequency of closed organ pipe of nth overtone is,
\(\mathrm{f}_{\mathrm{c}}=(\mathrm{p}+2) \frac{\mathrm{v}}{41}\)
The desired ratio is thus,
\(\frac{\mathrm{f}_0}{\mathrm{f}_{\mathrm{c}}}=\frac{2(\mathrm{p}+1)}{2 \mathrm{p}+1}\)

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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