MHT CET · Physics · Waves and Sound

On closing an open organ pipe from one end, it is noticed that the frequency of third harmonic is \(50 \mathrm{~Hz}\) more than the fundamental frequency of vibration in open organ pipe. The fundamental frequency of open organ pipe is

A \(100 \mathrm{~Hz}\)
B \(50 \mathrm{~Hz}\)
C \(250 \mathrm{~Hz}\)
D \(200 \mathrm{~Hz}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(100 \mathrm{~Hz}\)

Step-by-step Solution

Detailed explanation

Frequency of \(\mathrm{m}^{\text {th }}\) harmonic in an open organ pipe is \(f_m=\frac{m v}{2 L}\)
Fundamental frequency in an open organ pipe, for \(m=1 f_1=\frac{v}{2 L}\)
Frequency of nth harmonic in a closed organ pipe, \(f_n^{\prime}=\frac{n v}{4 L}\)
Frequency of 3rd harmonic in a closed organ pipe, \(f^{\prime}{ }_3=\frac{3 v}{4 L}\)
Now, \(f_3^{\prime}-f_1=50\)
\(\frac{3 v}{4 L}-\frac{v}{2 L}=50 \Rightarrow \frac{v}{L}=200\)
Thus, fundamental frequency in an open organ pipe,
\(f_1=\frac{v}{2 L}=\frac{200}{2} \mathrm{~Hz}=100 \mathrm{~Hz}\)

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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