Edexcel M2 (Mechanics 2) 2008 June

1. A lorry of mass 2000 kg is moving down a straight road inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 25 }\). The resistance to motion is modelled as a constant force of magnitude 1600 N . The lorry is moving at a constant speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

Find, in kW , the rate at which the lorry's engine is working.

2. A particle \(A\) of mass \(4 m\) is moving with speed \(3 u\) in a straight line on a smooth horizontal table. The particle \(A\) collides directly with a particle \(B\) of mass \(3 m\) moving with speed \(2 u\) in the same direction as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(e\). Immediately after the collision the speed of \(B\) is \(4 e u\).

1. Show that \(e = \frac { 3 } { 4 }\).
2. Find the total kinetic energy lost in the collision.

3. \begin{figure}[h] \end{figure} A package of mass 3.5 kg is sliding down a ramp. The package is modelled as a particle and the ramp as a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The package slides down a line of greatest slope of the plane from a point \(A\) to a point \(B\), where \(A B = 14 \mathrm {~m}\). At \(A\) the package has speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at \(B\) the package has speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Figure 1. Find

1. the total energy lost by the package in travelling from \(A\) to \(B\),
2. the coefficient of friction between the package and the ramp.

1. A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds,

$$\mathbf { F } = ( 6 t - 5 ) \mathbf { i } + \left( t ^ { 2 } - 2 t \right) \mathbf { j }$$ The velocity of \(P\) at time \(t\) seconds is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , \mathbf { v } = \mathbf { i } - 4 \mathbf { j }\).

1. Find \(\mathbf { v }\) at time \(t\) seconds. When \(t = 3\), the particle \(P\) receives an impulse ( \(- 5 \mathbf { i } + 12 \mathbf { j }\) ) N s.
2. Find the speed of \(P\) immediately after it receives the impulse.

5. \begin{figure}[h] \end{figure} A plank rests in equilibrium against a fixed horizontal pole. The plank is modelled as a uniform rod \(A B\) and the pole as a smooth horizontal peg perpendicular to the vertical plane containing \(A B\). The rod has length \(3 a\) and weight \(W\) and rests on the peg at \(C\), where \(A C = 2 a\). The end \(A\) of the rod rests on rough horizontal ground and \(A B\) makes an angle \(\alpha\) with the ground, as shown in Figure 2.

1. Show that the normal reaction on the rod at \(A\) is \(\frac { 1 } { 4 } \left( 4 - 3 \cos ^ { 2 } \alpha \right) W\). Given that the rod is in limiting equilibrium and that \(\cos \alpha = \frac { 2 } { 3 }\),
2. find the coefficient of friction between the rod and the ground.

6. \begin{figure}[h] \end{figure} Figure 3 shows a rectangular lamina \(O A B C\). The coordinates of \(O , A , B\) and \(C\) are ( 0,0 ), \(( 8,0 ) , ( 8,5 )\) and \(( 0,5 )\) respectively. Particles of mass \(k m , 5 m\) and \(3 m\) are attached to the lamina at \(A , B\) and \(C\) respectively. The \(x\)-coordinate of the centre of mass of the three particles without the lamina is 6.4.

1. Show that \(k = 7\). The lamina \(O A B C\) is uniform and has mass \(12 m\).
2. Find the coordinates of the centre of mass of the combined system consisting of the three particles and the lamina. The combined system is freely suspended from \(O\) and hangs at rest.
3. Find the angle between \(O C\) and the horizontal.

7. \begin{figure}[h] \end{figure} A ball is thrown from a point \(A\) at a target, which is on horizontal ground. The point \(A\) is 12 m above the point \(O\) on the ground. The ball is thrown from \(A\) with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) below the horizontal. The ball is modelled as a particle and the target as a point \(T\). The distance \(O T\) is 15 m . The ball misses the target and hits the ground at the point \(B\), where \(O T B\) is a straight line, as shown in Figure 4. Find

1. the time taken by the ball to travel from \(A\) to \(B\),
2. the distance \(T B\). The point \(X\) is on the path of the ball vertically above \(T\).
3. Find the speed of the ball at \(X\).