2
\includegraphics[max width=\textwidth, alt={}, center]{102e108b-2a36-4765-9990-78e2dd4276c0-2_583_785_676_680} Three horizontal forces of magnitudes \(15 \mathrm {~N} , 11 \mathrm {~N}\) and 13 N act on a particle \(P\) in the directions shown in the diagram. The angles \(\alpha\) and \(\beta\) are such that \(\sin \alpha = 0.28 , \cos \alpha = 0.96 , \sin \beta = 0.8\) and \(\cos \beta = 0.6\).

1. Show that the component, in the \(y\)-direction, of the resultant of the three forces is zero.
2. Find the magnitude of the resultant of the three forces.
3. State the direction of the resultant of the three forces.
   \includegraphics[max width=\textwidth, alt={}, center]{102e108b-2a36-4765-9990-78e2dd4276c0-2_348_711_1804_717} A block \(B\) of mass 0.4 kg and a particle \(P\) of mass 0.3 kg are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. \(B\) is in contact with the table and the part of the string between \(B\) and the pulley is horizontal. \(P\) hangs freely below the pulley (see diagram).
4. The system is in limiting equilibrium with the string taut and \(P\) on the point of moving downwards. Find the coefficient of friction between \(B\) and the table.
5. A horizontal force of magnitude \(X \mathrm {~N}\), acting directly away from the pulley, is now applied to \(B\). The system is again in limiting equilibrium with the string taut, and with \(P\) now on the point of moving upwards. Find the value of \(X\).