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\includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-12_439_1095_258_525} Two particles \(P\) and \(Q\) of masses 0.5 kg and \(m \mathrm {~kg}\) respectively are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the top of two inclined planes. The particles are initially at rest with \(P\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal and \(Q\) on a plane inclined at \(45 ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes. A force of magnitude 0.8 N is applied to \(P\) acting down the plane, causing \(P\) to move down the plane (see diagram).

1. It is given that \(m = 0.3\), and that the plane on which \(Q\) rests is smooth. Find the tension in the string.
2. It is given instead that the plane on which \(Q\) rests is rough, and that after each particle has moved a distance of 1 m , their speed is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done against friction in this part of the motion is 0.5 J . Use an energy method to find the value of \(m\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.