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An object is formed by attaching a thin uniform rod $P Q$ to a uniform rectangular lamina $A B C D$. The lamina has mass $m$, and $A B = D C = 6 a , B C = A D = 3 a$. The rod has mass $M$ and length $3 a$. The end $P$ of the rod is attached to the mid-point of $A B$. The rod is perpendicular to $A B$ and in the plane of the lamina (see diagram). Show that the moment of inertia of the object about a smooth horizontal axis $l _ { 1 }$, through $Q$ and perpendicular to the plane of the lamina, is $3 ( 8 m + M ) a ^ { 2 }$.

Show that the moment of inertia of the object about a smooth horizontal axis $l _ { 2 }$, through the mid-point of $P Q$ and perpendicular to the plane of the lamina, is $\frac { 3 } { 4 } ( 17 m + M ) a ^ { 2 }$.

Find expressions for the periods of small oscillations of the object about the axes $l _ { 1 }$ and $l _ { 2 }$, and verify that these periods are equal when $m = M$.

\hfill \mbox{\textit{CAIE FP2 2015 Q10 EITHER}}