4
\includegraphics[max width=\textwidth, alt={}, center]{4acdc4d7-01f0-496f-9390-de5f136bc5f9-3_360_1006_260_571} A non-uniform beam \(A B\) of length 4 m and mass 5 kg has its centre of mass at the point \(G\) of the beam where \(A G = 2.5 \mathrm {~m}\). The beam is freely suspended from its end \(A\) and is held in a horizontal position by means of a wire attached to the end \(B\). The wire makes an angle of \(20 ^ { \circ }\) with the vertical and the tension is \(T \mathrm {~N}\) (see diagram).

1. Calculate \(T\).
2. Calculate the magnitude and the direction of the force acting on the beam at \(A\).
   \includegraphics[max width=\textwidth, alt={}, center]{4acdc4d7-01f0-496f-9390-de5f136bc5f9-3_488_1095_1169_525} One end of a light inextensible string of length \(l\) is attached to the vertex of a smooth cone of semivertical angle \(45 ^ { \circ }\). The cone is fixed to the ground with its axis vertical. The other end of the string is attached to a particle of mass \(m\) which rotates in a horizontal circle in contact with the outer surface of the cone. The angular speed of the particle is \(\omega\) (see diagram). The tension in the string is \(T\) and the contact force between the cone and the particle is \(R\).
3. By resolving horizontally and vertically, find two equations involving \(T\) and \(R\) and hence show that \(T = \frac { 1 } { 2 } m \left( \sqrt { 2 } g + l \omega ^ { 2 } \right)\).
4. When the string has length 0.8 m , calculate the greatest value of \(\omega\) for which the particle remains in contact with the cone.