આકૃતિમાં દર્શાવ્યા પ્રમાણે એક \(\mathrm{R}\) ત્રિજ્યાના ગોળામાથી \(1\) ત્રિજ્યાનો ગોળો કાપી નાખવામાં આવેલ છે વધેલા ભાગનું દ્રવ્યમાનકેન્દ્ર \(G\) આગળ મળતું હોય તો \(R\) કયા સમીકરણ વડે મેળવી શકાય?

A \(\left(\mathrm{R}^{2}-\mathrm{R}+1\right)(2-\mathrm{R})=1\)
B \(\left(\mathrm{R}^{2}+\mathrm{R}-1\right)(2-\mathrm{R})=1\)
C \(\left(\mathrm{R}^{2}+\mathrm{R}+1\right)(2-\mathrm{R})=1\)
D \(\left(\mathrm{R}^{2}-\mathrm{R}-1\right)(2-\mathrm{R})=1\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\left(\mathrm{R}^{2}+\mathrm{R}+1\right)(2-\mathrm{R})=1\)

Step-by-step Solution

Detailed explanation

By concept of \(COM\) \(\mathrm{m}_{1} \mathrm{R}_{1}=\mathrm{m}_{2} \mathrm{R}_{2}\) Remaining mass \(\times(2-\mathrm{R})=\) cavity mass \(\times(\mathrm{R}-1)\)…

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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