5 Three forces act on a particle. The first force has magnitude \(P \mathrm {~N}\) and acts horizontally due east. The second force has magnitude 5 N and acts horizontally due west. The third force has magnitude \(2 P \mathrm {~N}\) and acts vertically upwards. The resultant of these three forces has magnitude 25 N .

1. Calculate \(P\) and the angle between the resultant and the vertical. The particle has mass 3 kg and rests on a rough horizontal table. The coefficient of friction between the particle and the table is 0.15 .
2. Find the acceleration of the particle, and state the direction in which it moves.
   \includegraphics[max width=\textwidth, alt={}, center]{c6bac5bf-960e-4c3d-b9fa-c52de66ba719-3_485_876_797_598} Two particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string which passes over a small smooth pulley at the top of a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) has mass 0.2 kg and is held at rest on the plane. \(Q\) has mass 0.2 kg and hangs freely. The string is taut (see diagram). The coefficient of friction between \(P\) and the plane is 0.4 . The particle \(P\) is released.
3. State the tension in the string before \(P\) is released, and find the tension in the string after \(P\) is released.
   \(Q\) strikes the floor and remains at rest. \(P\) continues to move up the plane for a further distance of 0.8 m before it comes to rest. \(P\) does not reach the pulley.
4. Find the speed of the particles immediately before \(Q\) strikes the floor.
5. Calculate the magnitude of the contact force exerted on \(P\) by the plane while \(P\) is in motion.