3 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held at the point \(A\), where \(O A\) makes an angle \(\theta\) with the downward vertical through \(O\), and with the string taut. The particle \(P\) is projected perpendicular to \(O A\) in an upwards direction with speed \(u\). It then starts to move along a circular path in a vertical plane. The string goes slack when \(P\) is at \(B\), where angle \(A O B\) is \(90 ^ { \circ }\) and the speed of \(P\) is \(\sqrt { \frac { 4 } { 5 } \mathrm { ag } }\).

1. Find the value of \(\sin \theta\).
2. Find, in terms of \(m\) and \(g\), the tension in the string when \(P\) is at \(A\).
   \includegraphics[max width=\textwidth, alt={}, center]{3b50dc98-781e-4399-8165-ad5e3065df4b-06_846_767_258_689} An object is formed from a solid hemisphere, of radius \(2 a\), and a solid cylinder, of radius \(a\) and height \(d\). The hemisphere and the cylinder are made of the same material. The cylinder is attached to the plane face of the hemisphere. The line \(O C\) forms a diameter of the base of the cylinder, where \(C\) is the centre of the plane face of the hemisphere and \(O\) is common to both circumferences (see diagram). Relative to axes through \(O\), parallel and perpendicular to \(O C\) as shown, the centre of mass of the object is ( \(\mathrm { x } , \mathrm { y }\) ).