A uniform sphere of mass 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. The force
of attraction between the remaining part of the sphere and the mass m will be

Options

A. `(7F)/(9)`
B. `(2F)/(3)`
C. `(4F)/(9)`
D. `(F)/(3)`

Answer

A

Explanation

The force of attraction between the complete sphere and mass m is
`F=(GmM)/((2R)^(2))=(GmM)/(4R^(2))" ".......(i)`
Mass of complete sphere is `M=(4 pi)/(3)R^(3)rho.`
Mass of the cut out portion is `m_(0)=(4 pi)/(3)(R/(2))^(3)rho.`
Thus, `m_(0)=(M)/(8)` . The distance between the centre of the cut out portion and mass m = 2R- `(R)/(2)=(3R)/(2).`
Hence the force of attraction between the cut out portion and mass m is
`f=(Gm_(0)m)/((3R//2)^(2))=(G(M//8)m)/(9R^(2)//4)=(GmM)/(4R^(2)) xx (2)/(9)`
Using (i), we get `f=(2F)/(9).` There fore, the force of attraction between the remaining part of the sphere
and mass m = F-f=F- `(2F)/(9)=(7F)/(9)`