Edexcel M2 (Mechanics 2) 2012 June

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]

A particle \(P\) moves in such a way that its velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) at time \(t\) seconds is given by $$\mathbf { v } = \left( 3 t ^ { 2 } - 1 \right) \mathbf { i } + \left( 4 t - t ^ { 2 } \right) \mathbf { j }$$

1. Find the magnitude of the acceleration of \(P\) when \(t = 1\) Given that, when \(t = 0\), the position vector of \(P\) is i metres,
2. find the position vector of \(P\) when \(t = 3\)

2. A particle \(P\) of mass \(3 m\) is moving with speed \(2 u\) in a straight line on a smooth horizontal plane. The particle \(P\) collides directly with a particle \(Q\) of mass \(4 m\) moving on the plane with speed \(u\) in the opposite direction to \(P\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the speed of \(Q\) immediately after the collision. Given that the direction of motion of \(P\) is reversed by the collision,
2. find the range of possible values of \(e\).

3. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of mass 5 kg and length 4 m , has its end \(A\) smoothly hinged at a fixed point. The rod is held in equilibrium at an angle of \(25 ^ { \circ }\) above the horizontal by a force of magnitude \(F\) newtons applied to its end \(B\). The force acts in the vertical plane containing the rod and in a direction which makes an angle of \(40 ^ { \circ }\) with the rod, as shown in Figure 1.

1. Find the value of \(F\).
2. Find the magnitude and direction of the vertical component of the force acting on the rod at \(A\).

4. \begin{figure}[h] \end{figure} A uniform circular disc has centre \(O\) and radius 4a. The lines \(P Q\) and \(S T\) are perpendicular diameters of the disc. A circular hole of radius \(2 a\) is made in the disc, with the centre of the hole at the point \(R\) on \(O P\) where \(O R = 2 a\), to form the lamina \(L\), shown shaded in Figure 2.

1. Show that the distance of the centre of mass of \(L\) from \(P\) is \(\frac { 14 a } { 3 }\). The mass of \(L\) is \(m\) and a particle of mass \(k m\) is now fixed to \(L\) at the point \(P\). The system is now suspended from the point \(S\) and hangs freely in equilibrium. The diameter \(S T\) makes an angle \(\alpha\) with the downward vertical through \(S\), where \(\tan \alpha = \frac { 5 } { 6 }\).
2. Find the value of \(k\).

5. \begin{figure}[h] \end{figure} A small ball \(B\) of mass 0.25 kg is moving in a straight line with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal plane when it is given an impulse. The impulse has magnitude 12.5 N s and is applied in a horizontal direction making an angle of \(\left( 90 ^ { \circ } + \alpha \right)\), where \(\tan \alpha = \frac { 3 } { 4 }\), with the initial direction of motion of the ball, as shown in Figure 3.

1. Find the speed of \(B\) immediately after the impulse is applied.
2. Find the direction of motion of \(B\) immediately after the impulse is applied.

6. A car of mass 1200 kg pulls a trailer of mass 400 kg up a straight road which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 14 }\). The trailer is attached to the car by a light inextensible towbar which is parallel to the road. The car's engine works at a constant rate of 60 kW . The non-gravitational resistances to motion are constant and of magnitude 1000 N on the car and 200 N on the trailer. At a given instant, the car is moving at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the acceleration of the car at this instant,
2. the tension in the towbar at this instant. The towbar breaks when the car is moving at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find, using the work-energy principle, the further distance that the trailer travels before coming instantaneously to rest.

7. \begin{figure}[h] \end{figure} A small stone is projected from a point \(O\) at the top of a vertical cliff \(O A\). The point \(O\) is 52.5 m above the sea. The stone rises to a maximum height of 10 m above the level of \(O\) before hitting the sea at the point \(B\), where \(A B = 50 \mathrm {~m}\), as shown in Figure 4. The stone is modelled as a particle moving freely under gravity.

1. Show that the vertical component of the velocity of projection of the stone is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the speed of projection.
3. Find the time after projection when the stone is moving parallel to \(O B\).