Edexcel M2 (Mechanics 2) 2015 June

1. A van of mass 900 kg is moving down a straight road that is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 30 }\). The resistance to motion of the van has constant magnitude 570 N . The engine of the van is working at a constant rate of 12.5 kW .

At the instant when the van is moving down the road at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the van is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the value of \(a\).

2. \begin{figure}[h] \end{figure} The uniform lamina \(O A B C D\), shown in Figure 1, is formed by removing the triangle \(O A D\) from the square \(A B C D\) with centre \(O\). The square has sides of length \(2 a\).

1. Show that the centre of mass of \(O A B C D\) is \(\frac { 2 } { 9 } a\) from \(O\). The mass of the lamina is \(M\). A particle of mass \(k M\) is attached to the lamina at \(D\) to form the system \(S\). The system \(S\) is freely suspended from \(A\) and hangs in equilibrium with \(A O\) vertical.
2. Find the value of \(k\).

1. A particle \(P\) of mass 0.75 kg is moving with velocity \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse \(( 6 \mathbf { i } + 6 \mathbf { j } ) \mathrm { Ns }\). The angle between the velocity of \(P\) before the impulse and the velocity of \(P\) after the impulse is \(\theta ^ { \circ }\).

Find

1. the value of \(\theta\),
2. the kinetic energy gained by \(P\) as a result of the impulse.

1. A ladder \(A B\), of weight \(W\) and length \(2 l\), has one end \(A\) resting on rough horizontal ground. The other end \(B\) rests against a rough vertical wall. The coefficient of friction between the ladder and the wall is \(\frac { 1 } { 3 }\). The coefficient of friction between the ladder and the ground is \(\mu\). Friction is limiting at both \(A\) and \(B\). The ladder is at an angle \(\theta\) to the ground, where \(\tan \theta = \frac { 5 } { 3 }\). The ladder is modelled as a uniform rod which lies in a vertical plane perpendicular to the wall.

Find the value of \(\mu\).

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 10 kg is projected from a point \(A\) up a line of greatest slope \(A B\) of a fixed rough plane. The plane is inclined at angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) and \(A B = 6.5 \mathrm {~m}\), as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\mu\). The work done against friction as \(P\) moves from \(A\) to \(B\) is 245 J .

1. Find the value of \(\mu\). The particle is projected from \(A\) with speed \(11.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By using the work-energy principle,
2. find the speed of the particle as it passes through \(B\).

1. A particle \(P\) moves on the positive \(x\)-axis. The velocity of \(P\) at time \(t\) seconds is \(\left( 2 t ^ { 2 } - 9 t + 4 \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , P\) is 15 m from the origin \(O\).

Find

1. the values of \(t\) when \(P\) is instantaneously at rest,
2. the acceleration of \(P\) when \(t = 5\)
3. the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 5\)

7. \begin{figure}[h] \end{figure} At time \(t = 0\), a particle is projected from a fixed point \(O\) on horizontal ground with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta ^ { \circ }\) to the horizontal. The particle moves freely under gravity and passes through the point \(A\) when \(t = 4 \mathrm {~s}\). As it passes through \(A\), the particle is moving upwards at \(20 ^ { \circ }\) to the horizontal with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Figure 3.

1. Find the value of \(u\) and the value of \(\theta\). At the point \(B\) on its path the particle is moving downwards at \(20 ^ { \circ }\) to the horizontal with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the time taken for the particle to move from \(A\) to \(B\). The particle reaches the ground at the point \(C\).
3. Find the distance \(O C\).

1. Three identical particles \(P , Q\) and \(R\), each of mass \(m\), lie in a straight line on a smooth horizontal plane with \(Q\) between \(P\) and \(R\). Particles \(P\) and \(Q\) are projected directly towards each other with speeds \(4 u\) and \(2 u\) respectively, and at the same time particle \(R\) is projected along the line away from \(Q\) with speed \(3 u\). The coefficient of restitution between each pair of particles is \(e\). After the collision between \(P\) and \(Q\) there is a collision between \(Q\) and \(R\).
   1. Show that \(e > \frac { 2 } { 3 }\)
   It is given that \(e = \frac { 3 } { 4 }\)
2. Show that there will not be a further collision between \(P\) and \(Q\).