\(m\) द्रव्यमान के एक पिण्ड को, लम्बाई \(L\) तथा अनुप्रस्थ काट क्षेत्रफल \(A\) द्रव्यमानरहित के तार के एक सिरे से लटकाते हैं। तार के पदार्थ का यंग प्रत्यास्थता गुणांक \(Y\) है। यदि द्रव्यमान को थोड़ा सा नीचे खींचकर छोड़ देते हैं तो ऊर्ध्व दिशा में इसके दोलन की आवृत्ति होगी।

A \(f =\frac{1}{2 \pi} \sqrt{\frac{ YA }{ mL }}\)
B \(f=\frac{1}{2 \pi} \sqrt{\frac{Y L}{m A}}\)
C \(f=\frac{1}{2 \pi} \sqrt{\frac{m A}{Y L}}\)
D \(f =\frac{1}{2 \pi} \sqrt{\frac{ mL }{ YA }}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(f =\frac{1}{2 \pi} \sqrt{\frac{ YA }{ mL }}\)

Step-by-step Solution

Detailed explanation

An elastic wire can be treated as a spring with \(K =\frac{ Y A }{\ell}\) \(T =2 \pi \sqrt{\frac{ m }{ k }}\) \(f=\frac{1}{2 \pi} \sqrt{\frac{k}{m}}=\frac{1}{2 \pi} \sqrt{\frac{Y A}{m \ell}}\)

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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