$($a$)$ The composite object shown consists of a homogeneous disc of mass $3$ kg and homogeneous slender rods $\backslash ($ A B $\backslash )$ and $\backslash ($ C D $\backslash ).$ Rod $\backslash ($ A B $\backslash )$ has a density of $\backslash (2\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}\backslash )$ and $\backslash (\backslash$operatorname$\{$rod$\}$ C D $\backslash )$ has a density of $\backslash (6\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}\backslash ).$

$($i$)$ If you require that the centre of mass of this composite object lie at the pivot $\backslash ($ O $\backslash ),$ i$.$e$.$ the origin of the given reference frame, show that the length of rod $\backslash ($ C D $\backslash )$ must be $1\mathrm{.}13$ m $.$

$($ii$)$ Hence determine the mass moment of inertia of the composite object about the axis passing through $\backslash ($ O $\backslash )$ perpendicular to the page.