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)=(piGrho_(0)R^(3))/6[1/((x-(R//2))^(2))-8/^(2))]hati,hatg=(2pixGrho_(0)R)/3 hati`

Explanation

From the principle of superposition field may be consider due to a solid sphere of (radius `R`) positive mass and solid sphere of (Radius `R//2`) having negative mass.

`(i) vec(g)=(Grho_(0)4/3piR^(3))/(x^(2))hati+(Grho_(0)4/3pi(R//2)^(3))/((x-R//2)^(2))`
`vec(g)=piGrho_(0)R^(3)[-4/(3x^(2))+1/(6(x-R//2)^(2))]hati`
`vec(g)=(piGrho_(0)R^(3))/6[1/((x-R//2)^(2))-8/(x^(2))]hati`
(ii) `vec(g)_(p)` =Field due to sphere `(R)` of positive mass +field due to shpere `(R//2)` of negative mass
`=4/3piGrho_(0)vec(PO)+4/3piRrho_(0)vec(O'P)`
`=4/3 piGrhovec(PO)+4/3piRrho_(o) vec(O,P)`
`vec(g)_(P)=4/3piGrho_(0)vec(O'O) 4/3piGPrho_(0)(vec(O,P)+vecPO)`
`vec(g)_(p)=4/3piGrho_(0)xxR/2(-hati)`
`vec(g)_(P)=2/3piGrho_(0)R(-hati)`