किसी एकसमान तार की अनुप्रस्थकाट का क्षेत्रफल \('A^{\prime}\) है। इससे बनाये गये एक लोलक का आवर्तकाल \(T\) है। इस लोलक के गोलक से एक अतिरिक्त \(M\) द्रव्यमान जोड़ देने से लोलक का आवर्तकाल परिवर्तित होकर \(T_{M}\) हो जाता है। यदि इस तार के पदार्थ का यंग गुणांक ' \(Y\) हो तो \(\frac{1}{Y}\) का मान होगा : \((g=\) गुरुत्वीय त्वरण \()\)

A \(\left[ {{{\left( {\frac{{{T_M}}}{T}} \right)}^2} - 1} \right]\frac{{Mg}}{A}\;\;\;\;\;\;\)
B \(\;\left[ {1 - {{\left( {\frac{{{T_M}}}{T}} \right)}^2}} \right]\frac{A}{{{M_g}}}\)
C \(\;\left[ {1 - {{\left( {\frac{T}{{{T_M}}}} \right)}^2}} \right]\frac{A}{{{M_g}}}\)
D \(\;\left[ {{{\left( {\frac{{{T_M}}}{T}} \right)}^2} - 1} \right]\frac{A}{{{M_g}}}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\;\left[ {{{\left( {\frac{{{T_M}}}{T}} \right)}^2} - 1} \right]\frac{A}{{{M_g}}}\)

Step-by-step Solution

Detailed explanation

As we know, time period, \(T=2 \pi \sqrt{\frac{\ell}{g}}\) \( When\, additional \,mass \mathrm\,{M}\) is added then \(\mathrm{T}_{\mathrm{M}}=2 \pi \sqrt{\frac{\ell+\Delta \ell}{\mathrm{g}}}\) \(T_{\frac{M}{T}}=\sqrt{\frac{\ell+\Delta \ell}{\ell}}\) or…

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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