Two thin uniform rods `A` (`M.L`) and `B` (`3M,3L`) are joined as shown. Find the `M.I.` about an axis passing through the center of mass of system of rods and perpendicular to the length.

Explanation

Locate the centre of mass taking origin at 0
`X_(CM)=(Mxx(L)/(2)+3M(L+(3L)/(2)))/(M+3M)=2L`
A is CM of system of rods

`I_(A)=I_(1)+I_(2)=[(ML^(2))/(12)+M(2L-(L)/(2))^(2)]+[(3M(3L)^(2))/(12)+3M((5L)/(2)-2L)^(2)]`
`=ML^(2)[{(1)/(12)+(9)/(4)}+{(27)/(12)+(3)/(4)}]`
`(16 ML^(2))/(3)`