લંબગત તરંગમાં એક જ સંમયે શૃંગ અને નજીકના ગર્ત વચ્ચેનું અંતર \(4\;cm\) છે અને તે સ્થાને શૃંગ અને નજીકના ગર્ત વચ્ચેનું અંતર \(1\;cm\) છે તે જ સ્થાને બીજું શૃંગ \(0.4\;s\) પછી બને છે તો માધ્યમમાં દોલન કરતાં કણની મહત્તમ ઝડપ કેટલી હશે?

A \(\frac{{3\pi }}{2}\,cm/s\)
B \(\frac{{5\pi }}{2}\,cm/s\)
C \(\frac{{\pi }}{2}\,cm/s\)
D \(2\pi \,cm/s\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{{5\pi }}{2}\,cm/s\)

Step-by-step Solution

Detailed explanation

Transverse wave equation \(y=a \sin (\omega t+k x)\) Next crest appears at time interval of \(0.4 \mathrm{s}\) So, the time period \(T=0.4\) \(\therefore\) Angular Frequency \(\omega=\frac{2 \pi}{T}=\frac{2 \pi}{0.4}\) \(\therefore\) Speed of the particle…

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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