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\includegraphics[max width=\textwidth, alt={}, center]{fdf004fa-3726-4726-a0b8-60030812d451-3_755_561_248_790} Coplanar forces of magnitudes \(250 \mathrm {~N} , 160 \mathrm {~N}\) and 370 N act at a point \(O\) in the directions shown in the diagram, where the angle \(\alpha\) is such that \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). Calculate the magnitude of the resultant of the three forces. Calculate also the angle that the resultant makes with the \(x\)-direction.
\(5 P\) and \(Q\) are fixed points on a line of greatest slope of an inclined plane. The point \(Q\) is at a height of 0.45 m above the level of \(P\). A particle of mass 0.3 kg moves upwards along the line \(P Q\).

1. Given that the plane is smooth and that the particle just reaches \(Q\), find the speed with which it passes through \(P\).
2. It is given instead that the plane is rough. The particle passes through \(P\) with the same speed as that found in part (i), and just reaches a point \(R\) which is between \(P\) and \(Q\). The work done against the frictional force in moving from \(P\) to \(R\) is 0.39 J . Find the potential energy gained by the particle in moving from \(P\) to \(R\) and hence find the height of \(R\) above the level of \(P\).