MHT CET · Physics · Waves and Sound

When open pipe is closed from one end then third overtone of closed pipe is higher in frequency by \(150 \mathrm{~Hz}\) than second overtone of open pipe. The fundamental frequency of open end pipe will be

A 300Hz
B 500Hz
C 200Hz
D 400Hz

Verified Solution

Answer & Solution

Correct Answer

(A) 300Hz

Step-by-step Solution

Detailed explanation

\(3^{\mathrm{rd}}\) overtone of closed pipe \(=7 \mathrm{n}_{\mathrm{oc}}=\frac{7 \mathrm{v}}{4 \ell}\)
\(2^{\text {nd }}\) overtone of closed pipe \(=3 \mathrm{n}_{00}=\frac{3 \mathrm{v}}{2 \ell}\)
Now, \(\frac{7 \mathrm{v}}{4 \ell}=\frac{3 \mathrm{v}}{2 \ell}+150\)
\(\frac{7 v}{4 \ell}-\frac{3 v}{2 \ell}=150\)
\(\frac{7 v-6 v}{4 \ell}=150\)
\(\frac{v}{4 \ell}=150\)
\(\begin{array}{ll}\therefore \quad \ell= & \frac{v}{4 \times 150} \\ \therefore \quad & n_{00}=\frac{v}{2 \ell}=\frac{v \times 4 \times 150}{2 \times v}=300 \mathrm{~Hz}\end{array}\)

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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