\begin{center}
\includegraphics[max width=\textwidth, alt={}]{3b311657-f609-4e8d-81e6-b0cbc7a8cbae-18_598_601_440_772}
\end{center}

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).\\
(i) 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 $3 ( 8 m + M ) a ^ { 2 }$.\\

(ii) 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 $\frac { 3 } { 4 } ( 17 m + M ) a ^ { 2 }$.\\

(iii) 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 2017 Q10 EITHER}}