आरेख में दर्शाए अनुसार द्रव्यमान \(M\) का कोई पिण्ड दो द्रव्यमानहीन कमानियों के बीच किसी चिकने आनत तल पर रखा है। कमानियों के मुक्त सिरे दढ़ सपोर्ट से जुड़े हैं। यदि प्रत्येक कमानी स्थिरांक \(k\) है, तो दिए गए पिण्ड के दोलन की आवत्ति होगी।

A \(\frac{1}{2 \pi} \sqrt{\frac{ k }{2 M }}\)
B \(\frac{1}{2 \pi} \sqrt{\frac{2 k }{ Mg \sin \alpha}}\)
C \(\frac{1}{2 \pi} \sqrt{\frac{2 k }{ M }}\)
D \(\frac{1}{2 \pi} \sqrt{\frac{ k }{ Mg \sin \alpha}}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{1}{2 \pi} \sqrt{\frac{2 k }{ M }}\)

Step-by-step Solution

Detailed explanation

\(K _{ eq }= K _{1}+ K _{2}= K + K =2 K\) \(T =2 \pi \sqrt{\frac{ m }{ K _{ eq }}}=2 \pi \sqrt{\frac{ m }{2 K }}\) \(f =\frac{1}{ T }=\frac{1}{2 \pi} \sqrt{\frac{2 K }{ m }}\)

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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