એક હલકો (નાનો) ગ્રહ એક મોટા (દળીય) તારાને ફરતે \(R\) ત્રિજ્યામાં \(T\) જેટલા પરિભમ્રણના આવર્તકાળ થી પરિભ્રમણ કરે છે. જો ગ્રહ અને તારા વચ્ચે પ્રવર્તંતું આકર્ષણબળ \(R^{-3 / 2}\) સમપ્રમાણ છે તો સાચો વિકલ્પ પસંદ કરો.

A \(\mathrm{T}^2 \propto \mathrm{R}^{5 / 2}\)
B \(T^2 \propto R^{7 / 2}\)
C \(T^2 \propto R^{3 / 2}\)
D \(T^2 \propto R^3\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\mathrm{T}^2 \propto \mathrm{R}^{5 / 2}\)

Step-by-step Solution

Detailed explanation

\(\mathrm{F}=\frac{\mathrm{GMm}}{\mathrm{R}^{3 / 2}}=\mathrm{m} \omega^2 \mathrm{R}\) \(\omega^2 \propto \frac{1}{\mathrm{R}^{5 / 2}} \quad \because \mathrm{T}=\frac{2 \pi}{\omega} \quad \text { so }\) \(\mathrm{T}^2 \propto \mathrm{R}^{5 / 2}\)

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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