From a solid sphere of mass `M` and radius `R`, a solid sphere of radius `R//2` is removed as shown. Find gravitational force on mass `m` as shown

Explanation

Mass of smaller solid `M'=M/(4/3piR^(3)) 4/3 pi(R//2)^(2)=M/8`
Force on `m` due to sphere of mass `M` and radius `R`
`F_(1)=(GMm)/((2R)^(2))=1/4(GMm)/(R^(2))`
Force on `m` due to sphere of mass `M/8` and radius `r/2`
`F_(2)=(GM/8m)/((2R-R//2)^(2))=(GMm//8)/(9R^(2)//4)=(GMm)/(18R^(2))`
Net force on `m`
`F=F_(1)-F_(2)=(GMm)/(R^(2))(1/4-1/18)`
`=(GMm)/(R^(2))((9-2)/36)`
`=7/36(GMm)/(R^(2))`, towards left