A sphere of radius `R` has its centre at the origin. It has a uniform mass density `rho_(0)` except that there is a spherical hole of radius `r=R//2` whose centre is at `x=R//2` as in fig (a) find gravitational field at points on the axis for `xgtR` (ii) Show that the gravitational field inside the hole is uniform, find its magnitude and direction.

Answer

`vec(g)=+(piGrhoR^(3))/(6)[(1)/((x-(R//2))^(2))-(8)/(x^(2))]hat(i), vec(g)=-(2piG rho_(o)R)/(3)hat(i)`

Explanation

From the principal of superpostion field may be consider due to a solid sphere of (Radius R) positive mass and solid sphere of (Radius R/2) having nagative mass.
(i) `vec(g)=-(Grho_(o)(4)/(3)piR^(2))/(x^(2))hat(i)+(Grho_(o)(4)/(3)pi(R//2)^(2))/((x-R//2)^(2))hat(i)`
`vec(g)=piGrho_(o)R^(3)[-(4)/(3x^(2))+(1)/(6(x-R//2)^(2))]hat(i) rArr vec(g) = (piGrho_(o)R^(3))/(6)[(1)/((x-R//2)^(2))-(8)/(x^(2))]hat(i)`
(ii) `vec(g)_(P)` = Field due to sphere (R ) of positive mass + field due to sphere (R/2) of negative mas
`=(4)/(3)piGrho_(o)vec(PO)+(4)/(3)piRrho_(o)vec(O'P)`
`=(4)/(3)piGrho_(o)(vec(O'P)+vec(PO))`
`vec(g)_(P)=(4)/(3)piGrho_(o)vec(O'O) rArr vec(g)_(P)=(4)/(3)piGrho_(o)xx(R )/(2)(-hati)`
`rArr vec(g)_(P)=(2)/(3)piGrho_(o)R(-hati)`