6 A block of mass 3 kg is initially at rest on a rough horizontal plane. A force of magnitude 6 N is applied to the block at an angle of \(\theta\) above the horizontal, where \(\cos \theta = \frac { 24 } { 25 }\). The force is applied for a period of 5 s , during which time the block moves a distance of 4.5 m .

1. Find the magnitude of the frictional force on the block.
2. Show that the coefficient of friction between the block and the plane is 0.165 , correct to 3 significant figures.
3. When the block has moved a distance of 4.5 m , the force of magnitude 6 N is removed and the block then decelerates to rest. Find the total time for which the block is in motion.
   \includegraphics[max width=\textwidth, alt={}, center]{cb1cc219-608f-4f11-ab2c-97cc8f0798c7-12_512_1097_258_523} Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 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 edge of a smooth plane. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 5 } . P\) lies on the plane and \(Q\) hangs vertically below the pulley at a height of 0.8 m above the floor (see diagram). The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane. \(P\) is released from rest and \(Q\) moves vertically downwards.
4. Find the tension in the string and the magnitude of the acceleration of the particles.
   \(Q\) hits the floor and does not bounce. It is given that \(P\) does not reach the pulley in the subsequent motion.
5. Find the time, from the instant at which \(P\) is released, for \(Q\) to reach the floor.
6. When \(Q\) hits the floor the string becomes slack. Find the time, from the instant at which \(P\) is released, for the string to become taut again.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.