A fixed sphere of radius `R` and uniform density `rho` has a spherical cavity of radius `R//2` such that the surface of the cavity passes through the centre of the sphere. A particle of mass `m` is located at the centre `(A)` of the velocity. Calculated
(a) The gravitational field at `A`.
(b) The velocity with which the particle strikes the centre `O` of the sphere. (Neglect earth's gravity)

Explanation

(a)
Mass of biger sphere`=rho4/3piR^(3)=M`
Mass of smaller sphere `rh0.4/3pi(R//2)^(3)=M/8`
Gravitational field due to big sphere at `A`
`E_(A)=(GM(R//2))/(R^(3))=(GM)/(2R^(2))`
Gravitational field due to small sphere at `A`
`E'_(A)=0`
`(E_(A))_("net")=(GM)/(2R^(2))=(Grho.4/3piR^(3))/(2R^(2))=2/3pirhoGR` (towards`O`)
(b) `F=m(E_(A))_(net)=ma`
`a=(2//3)pirhoGR`
`v^(2)=u^(2)+2as=0+2.(2//3)pirhoGR.R/2`
`v=sqrt((2piGrhoR^(2)//3)/3)`