Two particles A and B, of masses 3m kg and 2m 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 plane. The plane is inclined at an angle θ to the horizontal. A lies on the plane and B hangs vertically, 0.8 m above the floor, which is horizontal. The string between A and the pulley is parallel to a line of greatest slope of the plane (see diagram). Initially A and B are at rest.

1. Given that the plane is smooth, find the value of θ for which A remains at rest. [3]
2. It is given instead that the plane is rough, θ = 30° and the acceleration of A up the plane is 0.1 m s^-2.
3. Show that the coefficient of friction between A and the plane is \(\frac{1}{10}\sqrt{3}\). [5]
4. When B reaches the floor it comes to rest.
5. Find the length of time after B reaches the floor for which A is moving up the plane. [You may assume that A does not reach the pulley.] [4]