ચાર \(M\) દળના કણ \(a\) ત્રિજયાના ચોરસના શીરોબિંદુ પર છે.આકૃતિમાં દર્શાવ્યા પ્રમાણે આ કણને બીજા કણોના ગુરુત્વાકર્ષણ ક્ષેત્રને લીધે વર્તુળાકાર કક્ષામાં ભ્રમણ કરતા હોય તો તેનો વેગ કેટલો હશે?

A \(1.35\sqrt {\frac{{GM}}{a}} \)
B \(1.16\sqrt {\frac{{GM}}{a}} \)
C \(1.41\sqrt {\frac{{GM}}{a}} \)
D \(1.21\sqrt {\frac{{GM}}{a}} \)

Verified Solution

Answer & Solution

Correct Answer

(B) \(1.16\sqrt {\frac{{GM}}{a}} \)

Step-by-step Solution

Detailed explanation

Net force on particle towards center of circle is \({F_c} = \frac{{G{M^2}}}{{2{a^2}}} + \frac{{G{M^2}}}{{{a^2}}}\sqrt 2 \) \( = \frac{{G{M^2}}}{{{a^2}}}\left( {\frac{1}{2} + \sqrt 2 } \right)\) This force will act as centripetal force. Diatance of particle from center of circle…

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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