Edexcel M2 (Mechanics 2) 2014 June

1. A van of mass 600 kg is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 16 }\). The resistance to motion of the van from non-gravitational forces has constant magnitude \(R\) newtons. When the van is moving at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the van’s engine is working at a constant rate of 25 kW .
   1. Find the value of \(R\).
   The power developed by the van’s engine is now increased to 30 kW . The resistance to motion from non-gravitational forces is unchanged. At the instant when the van is moving up the road at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the van is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the value of \(a\).

1. A ball of mass 0.4 kg is moving in a horizontal plane when it is struck by a bat. The bat exerts an impulse \(( - 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { N }\) s on the ball. Immediately after receiving the impulse the ball has velocity \(( 12 \mathbf { i } + 15 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

Find

1. the speed of the ball immediately before the impact,
2. the size of the angle through which the direction of motion of the ball is deflected by the impact.

3. \begin{figure}[h] \end{figure} A non-uniform rod, \(A B\), of mass \(m\) and length 2l, rests in equilibrium with one end \(A\) on a rough horizontal floor and the other end \(B\) against a rough vertical wall. The rod is in a vertical plane perpendicular to the wall and makes an angle of \(60 ^ { \circ }\) with the floor as shown in Figure 1. The coefficient of friction between the rod and the floor is \(\frac { 1 } { 4 }\) and the coefficient of friction between the rod and the wall is \(\frac { 2 } { 3 }\). The rod is on the point of slipping at both ends.

1. Find the magnitude of the vertical component of the force exerted on the rod by the floor. The centre of mass of the rod is at \(G\).
2. Find the distance \(A G\).

4. \begin{figure}[h] \end{figure} Figure 2 shows a lamina \(L\). It is formed by removing a square \(P Q R S\) from a uniform triangle \(A B C\). The triangle \(A B C\) is isosceles with \(A C = B C\) and \(A B = 12 \mathrm {~cm}\). The midpoint of \(A B\) is \(D\) and \(D C = 8 \mathrm {~cm}\). The vertices \(P\) and \(Q\) of the square lie on \(A B\) and \(P Q = 4 \mathrm {~cm}\). The centre of the square is \(O\). The centre of mass of \(L\) is at \(G\).

1. Find the distance of \(G\) from \(A B\). When \(L\) is freely suspended from \(A\) and hangs in equilibrium, the line \(A B\) is inclined at \(25 ^ { \circ }\) to the vertical.
2. Find the distance of \(O\) from \(D C\).

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is released from rest at a point \(A\) on a rough inclined plane and slides down a line of greatest slope. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. The point \(B\) is 5 m from \(A\) on the line of greatest slope through \(A\), as shown in Figure 3.

1. Find the potential energy lost by \(P\) as it moves from \(A\) to \(B\). The speed of \(P\) as it reaches \(B\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. 1. Use the work-energy principle to find the magnitude of the constant frictional force acting on \(P\) as it moves from \(A\) to \(B\).
   2. Find the coefficient of friction between \(P\) and the plane. The particle \(P\) is now placed at \(A\) and projected down the plane towards \(B\) with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that the frictional force remains constant,
3. find the speed of \(P\) as it reaches \(B\).

6. \begin{figure}[h] \end{figure} A particle \(P\) is projected from a point \(A\) with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\alpha\), where \(\sin \alpha = \frac { 4 } { 5 }\). The point \(A\) is 10 m vertically above the point \(O\) which is on horizontal ground, as shown in Figure 4. The particle \(P\) moves freely under gravity and reaches the ground at the point \(B\). Calculate

1. the greatest height above the ground of \(P\), as it moves from \(A\) to \(B\),
2. the distance \(O B\). The point \(C\) lies on the path of \(P\). The direction of motion of \(P\) at \(C\) is perpendicular to the direction of motion of \(P\) at \(A\).
3. Find the time taken by \(P\) to move from \(A\) to \(C\).

7. A particle \(P\) of mass \(2 m\) is moving in a straight line with speed \(3 u\) on a smooth horizontal table. A second particle \(Q\) of mass \(3 m\) is moving in the opposite direction to \(P\) along the same straight line with speed \(u\). The particle \(P\) collides directly with \(Q\). The direction of motion of \(P\) is reversed by the collision. The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Show that the speed of \(Q\) immediately after the collision is \(\frac { u } { 5 } ( 8 e + 3 )\)
2. Find the range of possible values of \(e\). The total kinetic energy of the particles before the collision is \(T\). The total kinetic energy of the particles after the collision is \(k T\). Given that \(e = \frac { 1 } { 2 }\)
3. find the value of \(k\).
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