MHT CET · Physics · Waves and Sound

If ' \(l\) ' is the length of pipe, ' \(r\) ' is the internal radius of the pipe and ' \(v\) ' is the velocity of sound in air then fundamental frequency of open pipe is

A \(\frac{\mathrm{V}}{2(l+1 \cdot 2 \mathrm{r})}\)
B \(\frac{\mathrm{V}}{(l+1 \cdot 2 \mathrm{r})}\)
C \(\frac{\mathrm{V}}{(l+0 \cdot 3 \mathrm{r})}\)
D \(\frac{\mathrm{V}}{(l+0 \cdot 6 \mathrm{r})}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{\mathrm{V}}{2(l+1 \cdot 2 \mathrm{r})}\)

Step-by-step Solution

Detailed explanation

For an open organ pipe, the length of the pipe with end correction is given as:
\(\mathrm{L}=l+2 \mathrm{e}=l+2 \times 0.6 \mathrm{r} \ldots(\because \mathrm{e}=\) \(0.6 \mathrm{r} \text { for open pipe })\)
\(\mathrm{L}=l+1.2 \mathrm{r}\)
\(\therefore \quad\) The fundamental frequency of open pipe is:
\(\mathrm{f}=\frac{\mathrm{v}}{2 \mathrm{~L}}=\frac{\mathrm{v}}{2(l+1.2 \mathrm{r})}\)

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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