\(Y, K\) અને \(n\) એ કોઈ દ્રવ્ય માટે અનુક્રમે યંગ મોડયુલસ (યંગ માપાંક), બલ્ક મોડ્યુલસ (આયતન માપાંક) અને મોડયુલસ ઑફ રિજીડીટી (દઢતા માપાંક) નાં મૂલ્ય દર્શાવે છે. આ પ્રાચલો માટે સાચો સંબંધ પસંદ કરો.

A \(Y =\frac{9 K \eta}{3 K -\eta} N / m ^{2}\)
B \(\eta=\frac{3 YK }{9 K + Y } N / m ^{2}\)
C \(Y =\frac{9 K \eta}{2 \eta+3 K } N / m ^{2}\)
D \(K =\frac{ Y\eta }{9 \eta-3 Y } N / m ^{2}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(K =\frac{ Y\eta }{9 \eta-3 Y } N / m ^{2}\)

Step-by-step Solution

Detailed explanation

Y- Younge modulus, \(K\) - Bulk modulus, \(\eta\) - modulus of rigidity We know that \(y=3 k(1-2 \sigma)\) \(\sigma=\frac{1}{2}\left(1-\frac{ y }{3 k }\right)\) \(.........(i)\) \(y =2 \eta(1+\sigma)\) \(\sigma=\frac{y}{2 \eta}-1\) \(........(ii)\) From Eq.\((i)\) and Eq.…

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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