On to a sphere of radius `R//2` and density `P_(2)` with centre at `C_(2)` a second solid sphere is moulded with density `p_(1)` radius `R` and centre `C_(1)`. Find the force experienced by a point mass `m` at point `P` at a distance `y` from the combination as shown.

Explanation

If we consider that a sphere of radius `R` is placed with centre at `C_(1)` od density `rho_(1)` the force on the mass at `P` is
`F_(1)=G((4//3)piR^(3)rho_(1)m)/((R+y)^(2))` towards the sphere.
If we consider that a sphere of radius `R//2` is placed with centre at `C_(2)` of density `rho_(1)` the force on the mass at `P` is
`F_(2)=G((4//3)pi(R//2)^(3)rho_(1)m)/((R//2+R+y)^(2))` towards the sphere.
If we consider that a sphere of radius `R//2` is placed with centre at `C_(2)` of density `rho_(2)` the force on the mass in at `P`
`F_(3)=(G(4//3)pi(R//2)^(3)rho_(2)m)/((R//2+R+y)^(2))`
By the principle of superposition
`F=F_(1)-F_(2)+F_(3)=(4)/(3)piR^(3)Gm[(rho_(1))/((R+y)^(2))+((rho_(2)-rho_(1))//8)/(((3R//2)+y)^(2))]`