The density inside a solid sphere of radius `a` is given by `rho=rho_(0)(1-r/a)`, where `rho_(0)` is the density at the surface and `r` denotes the distance from the centre. Find the gravitational field due to this sphere at a distance `2a` from its centre.

Explanation

Volume of spherical element
`dV=4pir^(2) dr`
Mass of element
`dM=rhodV=rho_(0)(1-r/a)4pir^(2)dr`
Mass of sphere
`M=4pirho_(0)int_(0)^(a)(r^(2)-(r^(3))/a)dr`
`=4pirho|(r^(3))/3-(r^(4))/(4a)|_(0)^(a)`
`=(pirho_(0)a^(3))/3`

Gravitational field at `P`
`F=(GM)/((2a)^(2))=(Gpirho_(0)a^(3))/(12a^(2))`
`=1/12 piGrho_(0)a`