A uniform sphere of mas M and radius R exerts a force F on a small mass m situated at a distance of 2R from the centre O of the sphere. A spherical portion of diameter R is cut from the sphere as shown in Fig. 6.20. The force of attraction between the remaining part of the sphere and the mass m will be 

The force of attraction between the complete sphere and mass m is

                  $F=\frac{GmM}{{\left(2R\right)}^2}=\frac{GmM}{4R^2} \left(i\right)$

Mass of complete sphere is $M=\frac{4\mathrm{\pi }}{3}{R}^3\rho$.Thus $m_0=\frac{M}{8}$.The distance between the centre of the cut out portion an mass m = 2R$-\frac{R}{2}$=$\frac{3R}{2}$.

Hence the force of attraction between the cut out portion and mass m is 

           $f=\frac{G{m}_{0}m}{{\left(3R/2\right)}^2}=\frac{G\left(M/8\right)m}{{\displaystyle \raisebox{1ex}{$9R^2$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}=\frac{GmM}{4R^2}\times \frac{2}{9}$

Using (i), we get f = $\frac{2F}{9}$. Therefore, the force of attraction between the remaining part of the sphere and mass m = F$-$f= F$-$$\frac{2F}{9}$ = $\frac{7F}{9}$ ; which is choice(a).