Edexcel M2 (Mechanics 2) 2015 June

1. A particle of mass 0.3 kg is moving with velocity \(( 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(( - 3 \mathbf { i } + 3 \mathbf { j } ) \mathrm { N } \mathrm { s }\). Find the change in the kinetic energy of the particle due to the impulse.
   (6)
2. At time \(t\) seconds, \(t \geqslant 0\), a particle \(P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where

$$\mathbf { v } = \left( 27 - 3 t ^ { 2 } \right) \mathbf { i } + \left( 8 - t ^ { 3 } \right) \mathbf { j }$$ When \(t = 1\), the particle \(P\) is at the point with position vector \(\mathbf { r } \mathrm { m }\) relative to a fixed origin \(O\), where \(\mathbf { r } = - 5 \mathbf { i } + 2 \mathbf { j }\) Find

1. the magnitude of the acceleration of \(P\) at the instant when it is moving in the direction of the vector \(\mathbf { i }\),
2. the position vector of \(P\) at the instant when \(t = 3\)

1. A thin uniform wire of mass \(12 m\) is bent to form a right-angled triangle \(A B C\). The lengths of the sides \(A B , B C\) and \(A C\) are \(3 a , 4 a\) and \(5 a\) respectively. A particle of mass \(2 m\) is attached to the triangle at \(B\) and a particle of mass \(3 m\) is attached to the triangle at \(C\). The bent wire and the two particles form the system \(S\).

The system \(S\) is freely suspended from \(A\) and hangs in equilibrium.
Find the size of the angle between \(A B\) and the downward vertical.

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass 6.5 kg is projected up a fixed rough plane with initial speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(X\) on the plane, as shown in Figure 1. The particle moves up the plane along the line of greatest slope through \(X\) and comes to instantaneous rest at the point \(Y\), where \(X Y = d\) metres. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 3 } \cdot\)

1. Use the work-energy principle to show that, to 2 significant figures, \(d = 2.7\) After coming to rest at \(Y\), the particle \(P\) slides back down the plane.
2. Find the speed of \(P\) as it passes through \(X\).

1. Three particles \(A , B\) and \(C\) lie at rest in a straight line on a smooth horizontal table with \(B\) between \(A\) and \(C\). The masses of \(A , B\) and \(C\) are \(3 m\), 4m, and 5m respectively. Particle \(A\) is projected with speed \(u\) towards particle \(B\) and collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 3 }\).
   1. Show that the impulse exerted by \(A\) on \(B\) in this collision has magnitude \(\frac { 16 } { 7 } m u\)
   After the collision between \(A\) and \(B\) there is a direct collision between \(B\) and \(C\).
   After this collision between \(B\) and \(C\), the kinetic energy of \(C\) is \(\frac { 72 } { 245 } m u ^ { 2 }\)
2. Find the coefficient of restitution between \(B\) and \(C\).

6. \begin{figure}[h] \end{figure} A uniform \(\operatorname { rod } A B\) has length \(4 a\) and weight \(W\). A particle of weight \(k W , k < 1\), is attached to the rod at \(B\). The rod rests in equilibrium against a fixed smooth horizontal peg. The end \(A\) of the rod is on rough horizontal ground, as shown in Figure 2. The rod rests on the peg at \(C\), where \(A C = 3 a\), and makes an angle \(\alpha\) with the ground, where \(\tan \alpha = \frac { 1 } { 3 }\). The peg is perpendicular to the vertical plane containing \(A B\).

1. Give a reason why the force acting on the rod at \(C\) is perpendicular to the rod.
2. Show that the magnitude of the force acting on the rod at \(C\) is $$\frac { \sqrt { 10 } } { 5 } W ( 1 + 2 k )$$ The coefficient of friction between the rod and the ground is \(\frac { 3 } { 4 }\).
3. Show that for the rod to remain in equilibrium \(k \leqslant \frac { 2 } { 11 }\).

1. \hspace{0pt} [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.]

At time \(t = 0\), a particle \(P\) is projected with velocity ( \(4 \mathbf { i } + 9 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) from a fixed point \(O\) on horizontal ground. The particle moves freely under gravity. When \(P\) is at the point \(H\) on its path, \(P\) is at its greatest height above the ground.

1. Find the time taken by \(P\) to reach \(H\). At the point \(A\) on its path, the position vector of \(P\) relative to \(O\) is \(( k \mathbf { i } + k \mathbf { j } ) \mathrm { m }\), where \(k\) is a positive constant.
2. Find the value of \(k\).
   (4)
3. Find, in terms of \(k\), the position vector of the other point on the path of \(P\) which is at the same vertical height above the ground as the point \(A\).
   (3) At time \(T\) seconds the particle is at the point \(B\) and is moving perpendicular to \(( 4 \mathbf { i } + 9 \mathbf { j } )\)
4. Find the value of \(T\).