3 The region bounded by the \(x\)-axis, the lines \(x = 1\) and \(x = 2\) and the curve \(y = \frac { 1 } { x ^ { 2 } }\) for \(1 \leqslant x \leqslant 2\), is occupied by a uniform lamina of mass 24 kg . The unit of length is the metre. Find the moment of inertia of this lamina about the \(x\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{d5c6deb0-ef1a-4878-889d-dc9f926aaf88-2_623_601_1409_706} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is freely hinged to a fixed point at \(A\). A particle of mass \(2 m\) is attached to the rod at \(B\). A light elastic string, with natural length \(a\) and modulus of elasticity \(5 m g\), passes through a fixed smooth ring \(R\). One end of the string is fixed to \(A\) and the other end is fixed to the mid-point \(C\) of \(A B\). The ring \(R\) is at the same horizontal level as \(A\), and is at a distance \(a\) from \(A\). The rod \(A B\) and the ring \(R\) are in a vertical plane, and \(R C\) is at an angle \(\theta\) above the horizontal, where \(0 < \theta < \frac { 1 } { 4 } \pi\), so that the acute angle between \(A B\) and the horizontal is \(2 \theta\) (see diagram).

1. By considering the energy of the system, find the value of \(\theta\) for which the system is in equilibrium.
2. Determine whether this position of equilibrium is stable or unstable.