\(m\) દળ ધરાવતો એક ઉપગ્રહ, પૃથ્વી (ત્રિજયા \(R\)) ની આસપાસ સપાટીથી \(h\) ઊંચાઇએ પરિક્રમણ કરે છે. જો પૃથ્વીની સપાટી પર ગુરુત્વપ્રવેગ \(g_0\) હોય, તો ઉપગ્રહની કુલઊર્જા \(g_0\) ના પદમાં કઇ હશે?

A \(\frac{{2m{g_0}{R^2}}}{{R + h}}\)
B \(-\)\(\;\frac{{2m{g_0}{R^2}}}{{R + h}}\)
C \(\;\frac{{{mg_0}{R^2}}}{{2\left( {R + h} \right)}}\)
D \(-\)\(\;\frac{{{mg_0}{R^2}}}{{2\left( {R + h} \right)}}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(-\)\(\;\frac{{{mg_0}{R^2}}}{{2\left( {R + h} \right)}}\)

Step-by-step Solution

Detailed explanation

Total energy of satellite at height \(h\) from the earth surface, \(E=PE+KE\)

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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