A solid sphere of radius \(R\) gravitationally attracts a particle placed at \(3 R\) form its centre with a force \(F _{1}\). Now a spherical cavity of radius \(\left(\frac{ R }{2}\right)\) is made in the sphere (as shown in figure) and the force becomes \(F _{2}\). The value of \(F _{1}: F _{2}\) is

\(F _{1}= m . E .=\frac{ m \cdot Gm ^{\prime}}{9 R ^{2}}\)

\(F_{2}=m\left[\frac{G \cdot m^{\prime}}{(3 R)^{2}}-\frac{G \cdot m^{\prime} / 8}{(5 R / 2)^{2}}\right]\)

\(=\frac{G m^{\prime}}{9 R^{2}}-\frac{G m^{\prime} \times 4}{8 \times 25}\)

\(=\left(\frac{1}{9}-\frac{1}{50}\right) \frac{G m^{\prime}}{R^{2}}\)

\(F_{2}=\frac{41}{50 \times 9} \cdot \frac{G m^{\prime}}{R^{2}}\)

\(\frac{F_{1}}{F_{2}}=\frac{1}{9} \times \frac{50 \times 9}{41}=\frac{50}{41}\)