આકૃતિમાં દર્શાવ્યા પ્રમાણે એક \(b\) ત્રિજ્યા ધરાવતી તક્તી માં \(a\) ત્રિજ્યા ધરાવતો હૉલ છે.જો તક્તીનું દળ પ્રતિ ક્ષેત્રફળ \(\left( {\frac{{{\sigma _0}}}{r}} \right)\)મુજબ બદલાતું હોય તો તક્તી કેન્દ્રમાથી પસાર થતી અક્ષને અનુલક્ષીને ચક્રાવર્તન ત્રિજ્યા કેટલી થાય?

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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