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A uniform disc, with centre $O$ and radius $a$, is surrounded by a uniform concentric ring with radius $3 a$. The ring is rigidly attached to the rim of the disc by four symmetrically positioned uniform rods, each of mass $\frac { 3 } { 2 } m$ and length $2 a$. The disc and the ring each have mass $2 m$. The rods meet the ring at the points $A , B , C$ and $D$. The disc, the ring and the rods are all in the same plane (see diagram). Show that the moment of inertia of this object about an axis through $O$ perpendicular to the plane of the object is $45 m a ^ { 2 }$.

Find the moment of inertia of the object about an axis $l$ through $A$ in the plane of the object and tangential to the ring.

A particle of mass $3 m$ is now attached to the object at $C$. The object, including the additional particle, is suspended from the point $A$ and hangs in equilibrium. It is free to rotate about the axis $l$. The centre of the disc is given a horizontal speed $u$. When, in the subsequent motion, the object comes to instantaneous rest, $C$ is below the level of $A$ and $A C$ makes an angle $\sin ^ { - 1 } \left( \frac { 1 } { 4 } \right)$ with the horizontal. Find $u$ in terms of $a$ and $g$.

\hfill \mbox{\textit{CAIE FP2 2015 Q11 EITHER}}