1.	A particle of mass 0.8 kg is held at rest on a rough plane. The plane is inclined at 30° to the horizontal. The particle is released from rest and slides down a line of greatest slope of the plane. The particle moves 2.7 m during the first 3 seconds of its motion. Find
(a)	the acceleration of the particle,
(3)
 
(b)	the coefficient of friction between the particle and the plane.
(5)
 
	The particle is now held on the same rough plane by a horizontal force of magnitude x newtons, acting in a plane containing a line of greatest slope of the plane, as shown in the diagram above. The particle is in equilibrium and on the point of moving up the plane.
		
(c)	Find the value of X.
(7)
(Total 15 marks)

2.	A small brick of mass 0.5 kg is placed on a rough plane which is inclined to the horizontal at an angle θ, where tan θ = , and released from rest. The coefficient of friction between the brick and the plane is .
	Find the acceleration of the brick.
(Total 9 marks)

3.	Two forces, (4i – 5j) N and (pi + qj) N, act on a particle P of mass m kg. The resultant of the two forces is R. Given that R acts in a direction which is parallel to the vector (i – 2j),
(a)	find the angle between R and the vector j,
(3)

(b)	show that 2p + q + 3 = 0.
(4)
 
	Given also that q = 1 and that P moves with an acceleration of magnitude 8√5 m s–2,
(c)	find the value of m.
(7)
(Total 14 marks)

4.	

	A particle P of mass 0.5 kg is on a rough plane inclined at an angle  to the horizontal,
where . The particle is held at rest on the plane by the action of a force of
magnitude 4 N acting up the plane in a direction parallel to a line of greatest slope of the
plane, as shown in the figure above. The particle is on the point of slipping up the plane.
(a)	Find the coefficient of friction between P and the plane.
(7)
 
The force of magnitude 4 N is removed.
(b)	Find the acceleration of P down the plane.
(4)
(Total 11 marks)

5.	

	A parcel of weight 10N lies on a rough plane inclined at an angle of 30° to the horizontal. A horizontal force of magnitude P newtons acts on the parcel, as shown in the figure above. The parcel is in equilibrium and on the point of slipping up the plane. The normal reaction of the plane on the parcel is 18N. The coefficient of friction between the parcel and the plane is m. Find
(a)	the value of P,
(4)
(b)	the value of m.
(5)
 
The horizontal force is removed.
(c)	Determine whether or not the parcel moves.
(5)
(Total 14 marks)

6.	

	A box of mass 2 kg is pulled up a rough plane face by means of a light rope. The plane is inclined at an angle of 20° to the horizontal, as shown in the diagram. The rope is parallel to a line of greatest slope of the plane. The tension in the rope is 18 N. The coefficient of friction between the box and the plane is 0.6. By modelling the box as a particle, find
(a)	the normal reaction of the plane on the box,
(3)

(b)	the acceleration of the box.
(5)
(Total 8 marks)

7.	

	The diagram above shows a boat B of mass 400 kg held at rest on a slipway by a rope. The boat is modelled as a particle and the slipway as a rough plane inclined at 15° to the horizontal. The coefficient of friction between B and the slipway is 0.2. The rope is modelled as a light, inextensible string, parallel to a line of greatest slope of the plane. The boat is in equilibrium and on the point of sliding down the slipway.
(a)	Calculate the tension in the rope.
(6)
 
	The boat is 50 m from the bottom of the slipway. The rope is detached from the boat and the boat slides down the slipway.
(b)	Calculate the time taken for the boat to slide to the bottom of the slipway.
(6)
(Total 12 marks)

8.	A tile on a roof becomes loose and slides from rest down the roof. The roof is modelled as a plane surface inclined at 30° to the horizontal. The coefficient of friction between the tile and the roof is 0.4. The tile is modelled as a particle of mass m kg.
 
(a)	Find the acceleration of the tile as it slides down the roof.
(7)

	The tile moves a distance 3 m before reaching the edge of the roof.
(b)	Find the speed of the tile as it reaches the edge of the roof.
(2)
 
(c)	Write down the answer to part (a) if the tile had mass 2m kg.
(1)
(Total 10 marks)

9.	A particle P of mass 3 kg is projected up a line of greatest slope of a rough plane inclined at an angle of 30° to the horizontal. The coefficient of friction between P and the plane is 0.4. The initial speed of P is 6 m s-1. Find
(a)	the frictional force acting on P as it moves up the plane,
(4)
(b)	the distance moved by P up the plane before P comes to instantaneous rest.
(7)
(Total 11 marks)

10.	

	A small parcel of mass 2 kg moves on a rough plane inclined at an angle of 30° to the horizontal. The parcel is pulled up a line of greatest slope of the plane by means of a light rope which is attached to it. The rope makes an angle of 30° with the plane, as shown in the diagram above. The coefficient of friction between the parcel and the plane is 0.4. 
	Given that the tension in the rope is 24 N,

(a)	find, to 2 significant figures,  the acceleration of the parcel.
(8)
	The rope now breaks. The parcel slows down and comes to rest.
(b)	Show that, when the parcel comes to this position of rest, it immediately starts to move down the plane again.
(4)
(c)	Find, to 2 significant figures, the acceleration of the parcel as it moves down the plane after it has come to this position of instantaneous rest.
(3)
(Total 15 marks)