3 A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 2 m and modulus of elasticity \(2 m g \mathrm {~N}\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) hangs in equilibrium vertically below \(O\). The particle \(P\) is pulled down vertically a distance \(d\) m below its equilibrium position and released from rest.

1. Given that the particle just reaches \(O\) in the subsequent motion, find the value of \(d\).
   \includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-04_2717_33_109_2014}
   \includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-05_2725_35_99_20}
2. Hence find the speed of \(P\) when it is 2 m below \(O\).
   \includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-06_785_729_255_708} An object is formed by removing a cylinder of radius \(\frac { 2 } { 3 } a\) and height \(k h ( k < 1 )\) from a uniform solid cylinder of radius \(a\) and height \(h\). The vertical axes of symmetry of the two cylinders coincide. The upper faces of the two cylinders are in the same plane as each other. The points \(A\) and \(B\) are the opposite ends of a diameter of the upper face of the object (see diagram).
3. Find, in terms of \(h\) and \(k\), the distance of the centre of mass of the object from \(A B\).
   
   When the object is suspended from \(A\), the angle between \(A B\) and the vertical is \(\theta\), where \(\tan \theta = \frac { 3 } { 2 }\).
4. Given that \(h = \frac { 8 } { 3 } a\), find the possible values of \(k\).