Two uniform rods AB and BC are rigidly joined at B so that 
ABC = 90°. Rod AB has length 0.5 m and mass 2 kg. Rod BC has length 2 m and mass 3 kg. The centre of mass of the framework of the two rods is at G.
The distance of G from AB is 0.6 m. The framework is suspended from A and hangs freely in equilibrium.
Find the angle between AB and the downward vertical at A.
(3)
A lorry of mass 1800 kg travels along a straight horizontal road. The lorry’s engine is working at a constant rate of 30 kW. When the lorry’s speed is 20 m s−1, its acceleration is 0.4 m s−2. The magnitude of the resistance to the motion of the lorry is R newtons.
The lorry now travels up a straight road which is inclined at an angle 
to the horizontal, where sin α = 
. The magnitude of the non-gravitational resistance to motion is R newtons. The lorry travels at a constant speed of 20 m s−1.
Find the new rate of working of the lorry’s engine.
(5)
A ladder, of length 5 m and mass 18 kg, has one end A resting on rough horizontal ground and its other end B resting against a smooth vertical wall. The ladder lies in a vertical plane perpendicular to the wall and makes an angle 
with the horizontal ground, where tan α = 
, as shown in Figure 1. The coefficient of friction between the ladder and the ground is μ. A woman of mass 60 kg stands on the ladder at the point C, where AC = 3 m. The ladder is on the point of slipping. The ladder is modelled as a uniform rod and the woman as a particle. Find the value of μ.
(9)
At time t seconds the velocity of a particle P is [(4t – 5)i + 3j] m s–1. When t = 0, the position vector of P is (2 i + 5 j) m, relative to a fixed origin O.
Find the value of t when the velocity of P is parallel to the vector j.
(1)
Find an expression for the position vector of P at time t seconds.
(4)
A second particle Q moves with constant velocity (−2i + cj) m s−1. When t = 0, the position vector of Q is (11 i + 2 j) m. The particles P and Q collide at the point with position vector (d i +14 m. j)
Find
the value of c,
the value of d.
(5)
The point A lies on a rough plane inclined at an angle 
to the horizontal, where sin θ = 
A particle P is projected from A, up a line of greatest slope of the plane, with speed U m s−1. The mass of P is 2 kg and the coefficient of friction between P and the plane is 
. The particle comes to instantaneous rest at the point B on the plane, where AB =1 5. m. It then moves back down the plane to A.
Find the work done against friction as P moves from A to B.
(4)
Use the work-energy principle to find the value of U.
(4)
Find the speed of P when it returns to A.
(3)
A ball is thrown from a point O, which is 6 m above horizontal ground. The ball is projected with speed u m s−1 at an angle 
above the horizontal. There is a thin vertical post which is 4 m high and 8 m horizontally away from the vertical through O, as shown in Figure 2. The ball passes just above the top of the post 2 s after projection. The ball is modelled as a particle.
The ball hits the ground T seconds after projection.
Find the value of T.
(3)
Immediately before the ball hits the ground the direction of motion of the ball makes an angle 
with the horizontal.
(5)
A particle A of mass m is moving with speed u on a smooth horizontal floor when it collides directly with another particle B, of mass 3 m, which is at rest on the floor. The coefficient of restitution between the particles is e. The direction of motion of A is reversed by the collision.
Find, in terms of e and u,
the speed of A immediately after the collision,
the speed of B immediately after the collision.
(7)
After being struck by A the particle B collides directly with another particle C, of mass 4 m, which is at rest on the floor. The coefficient of restitution between B and C is 2 e. Given that the direction of motion of B is reversed by this collision,
find the range of possible values of e,
(6)
determine whether there will be a second collision between A and B.
(3)
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