\(R\) ત્રિજ્યા ધરાવતો એક ઘન ગોળો તેના કેન્દ્રથી \(3R\) અંતરે રહેલા કણને \(F_1\) જેટલા બળ દ્વારા ગુરુત્વાકર્ષણ બળ થી આકર્ષે છે. હવે, ગોળામાં \(\left(\frac{R}{2}\right)\) જેટલી ગોલીય બખોલ (છિદ્ર) કરવામાં આવે છે (આકૃત્તિમાં દર્શાવ્યા મુજબ) અને બળ \(F_2\) થાય છે. \(F _{1}: F _{2}\) નું મૂલ્ય ........... છે.

A \(25: 36\)
B \(36: 25\)
C \(50: 41\)
D \(41: 50\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(50: 41\)

Step-by-step Solution

Detailed explanation

Let initial mass of sphere is \(m ^{\prime} .\) Hence mass of removed portion will be \(m ^{\prime} / 8\) \(F _{1}= m . E .=\frac{ m \cdot Gm ^{\prime}}{9 R ^{2}}\) \(F_{2}=m\left[\frac{G \cdot m^{\prime}}{(3 R)^{2}}-\frac{G \cdot m^{\prime} / 8}{(5 R / 2)^{2}}\right]\)…

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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