From a thin uniform disc of radius `2R`. Another disc of diameter `2R` is removed. The mass of the remaining portion is `m`. Find the `M.I.` of the shaded portion about an axis passing through `O` and pependicular to the plane.

Explanation

Mass of disc `prop` area
`m_(A)proppi(2R)^(2)impliesm_(A)=4M`
`m_(B)proppiR^(2)impliesm_(B)=M`
`(I_(A))_(0)=(1)/(2)xx4M(2R)^(2)=8MR^(2)`
`(I_(B))_(0)=((1)/(2)xxMR^(2)+MR^(2))=(3MR^(2))/(2)`
`M.I.` of the shaded portion `I_(0)=(I_(A))_(0)-(I_(B))_(0)=(13MR^(2))/(2)`
`4M-M = m implies M = (m)/(3)`
`I_(0)=(13MR^(2))/(2)=(13mR^(2))/(6)`