આકૃતિમાં દર્શાવેલ ગોળા અને વલય (રીંગ) વચ્ચે આકર્ષણ બળ શોધો, જ્યાં રીંગનું સમતલ કેન્દ્રોને જોડતી રેખાને લંબરૂપે છે. બે રિંગ \(('m'\) દળ) નાં કેન્દ્ર થી ગોળા \(('M'\) દળ)નાં કેન્દ્ર વચ્ચેનું અંતર \(\sqrt{8} R\) હોય તેમજ બંને એકસરખી ત્રિજ્યા \('R’\) ધરાવે છે.

A \(\frac{\sqrt{8}}{9} \cdot \frac{ GmM }{ R }\)
B \(\frac{2 \sqrt{2}}{3} \cdot \frac{ GMm }{ R ^{2}}\)
C \(\frac{1}{3 \sqrt{8}} \cdot \frac{ GMm }{ R ^{2}}\)
D \(\frac{\sqrt{8}}{27} \cdot \frac{ GmM }{ R ^{2}}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{\sqrt{8}}{27} \cdot \frac{ GmM }{ R ^{2}}\)

Step-by-step Solution

Detailed explanation

Gravitational field of ring \(=-\frac{\text { Gmx }}{\left(R^{2}+x^{2}\right)^{3 / 2}}\) Force between sphere and ring \(=\frac{\operatorname{GmM}(\sqrt{8} R )}{\left( R ^{2}+8 R ^{2}\right)^{3 / 2}}\) \(=\frac{ GmM }{ R ^{2}} \times \frac{\sqrt{8}}{27}\)

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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