A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F=4 a\) and \(F G=2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B=6 a\) and \(B C=4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac{11}{12} m\) is fixed to the mid-point of \(A B\) (see diagram).
(i) Show that the moment of inertia of the system, consisting of frame and small object, about an axis through \(O\) perpendicular to the plane of the frame, is \(\frac{169}{3} m a^{2}\).
(ii) Show that small oscillations of the system about this axis are approximately simple harmonic and state their period.