किसी वृत्ताकार डिस्क की त्रिज्या \(b\) हैं इसमें एक छिद्र इसके केन्द्र पर बना है, जिसकी त्रिज्या \(a\) है, चित्र देखिए। यदि डिस्क के प्रति-एकांक-क्षेत्रफल का द्रव्यमान, \(\left(\frac{\sigma_{0}}{ r }\right)\) के अनुसार परिवर्तित होता हैं तो, इसके केन्द्र से गुजरने वाली अक्ष के परितः डिस्क की परिभ्रमण त्रिज्या होगी।

A \(\frac{{a + b}}{3}\)
B \(\sqrt {\frac{{{a^2} + {b^2} + ab}}{3}} \)
C \(\frac{{a + b}}{2}\)
D \(\sqrt {\frac{{{a^2} + {b^2} + ab}}{2}} \)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\sqrt {\frac{{{a^2} + {b^2} + ab}}{3}} \)

Step-by-step Solution

Detailed explanation

\(dI = \left( {dm} \right){r^2}\) \( = \left( {\sigma dA} \right){r^2}\) \( = \left( {\frac{{{\sigma _0}}}{r}2\pi dr} \right){r^2} = \left( {{\sigma _0}2\pi 0{r^2}dr} \right)\) \(I = \int {DI = \int\limits_a^b {{\sigma _0}2\pi {r^2}dr} } \)…

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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