બે સમાન દળવાળા કણ બળ \(F(r) = \frac{{ - 16}}{r}\, - \,{r^3}\) ના લીધે વર્તુળાકાર કક્ષામાં ગતિ કરે છે. પ્રથમ કણ \(r = 1\) અંતરે અને બીજો કણ \(r = 4\) અંતરે છે. તો પ્રથમ અને બીજા કણની ગતિ ઉર્જા નો અનુમાનિત ગુણોત્તર નીચે પૈકી શેની સૌથી નજીક મળે?

A \(10^{-1}\)
B \(6 \times {10^{-2}}\)
C \(6 \times {10^2}\)
D \(3 \times {10^{-3}}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(6 \times {10^{-2}}\)

Step-by-step Solution

Detailed explanation

As the particles moving in circular orbits, So \(\frac{{m{v^2}}}{r} = \frac{{16}}{r} + {r^3}\) Kinetic energy, \(K{E_0} = \frac{1}{2}m{v^2} = \frac{1}{2}\left[ {16 + {r^4}} \right]\) For fist particle, \(r = 1,\) \({K_1} = \frac{1}{2}\left( {16 + 1} \right)\) Similarly, for…

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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