Edexcel M2 (Mechanics 2) 2011 June

1. A car of mass 1000 kg moves with constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 30 }\). The engine of the car is working at a rate of 12 kW . The resistance to motion from non-gravitational forces has magnitude 500 N . Find the value of \(V\).
   
2. A particle \(P\) of mass \(m\) is moving in a straight line on a smooth horizontal surface with speed \(4 u\). The particle \(P\) collides directly with a particle \(Q\) of mass \(3 m\) which is at rest on the surface. The coefficient of restitution between \(P\) and \(Q\) is \(e\). The direction of motion of \(P\) is reversed by the collision.

Show that \(e > \frac { 1 } { 3 }\).

3. A ball of mass 0.5 kg is moving with velocity \(12 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is struck by a bat. The impulse received by the ball is \(( - 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { N } \mathrm { s }\). By modelling the ball as a particle, find

1. the speed of the ball immediately after the impact,
2. the angle, in degrees, between the velocity of the ball immediately after the impact and the vector \(\mathbf { i }\),
3. the kinetic energy gained by the ball as a result of the impact.

4. \begin{figure}[h] \end{figure} Figure 1 shows a uniform lamina \(A B C D E\) such that \(A B D E\) is a rectangle, \(B C = C D\), \(A B = 4 a\) and \(A E = 2 a\). The point \(F\) is the midpoint of \(B D\) and \(F C = a\).

1. Find, in terms of \(a\), the distance of the centre of mass of the lamina from \(A E\). The lamina is freely suspended from \(A\) and hangs in equilibrium.
2. Find the angle between \(A B\) and the downward vertical.

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is projected from a point \(A\) up a line of greatest slope \(A B\) of a fixed plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal and \(A B = 2 \mathrm {~m}\) with \(B\) above \(A\), as shown in Figure 2. The particle \(P\) passes through \(B\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The plane is smooth from \(A\) to \(B\).

1. Find the speed of projection. The particle \(P\) comes to instantaneous rest at the point \(C\) on the plane, where \(C\) is above \(B\) and \(B C = 1.5 \mathrm {~m}\). From \(B\) to \(C\) the plane is rough and the coefficient of friction between \(P\) and the plane is \(\mu\). By using the work-energy principle,
2. find the value of \(\mu\).

1. A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\) at time \(t\) seconds is \(( t - 4 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the positive \(x\)-direction. The velocity of \(P\) at time \(t\) seconds is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 0 , v = 6\).

Find

1. \(v\) in terms of \(t\),
2. the values of \(t\) when \(P\) is instantaneously at rest,
3. the distance between the two points at which \(P\) is instantaneously at rest.

7. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of mass \(3 m\) and length \(4 a\), is held in a horizontal position with the end \(A\) against a rough vertical wall. One end of a light inextensible string \(B D\) is attached to the rod at \(B\) and the other end of the string is attached to the wall at the point \(D\) vertically above \(A\), where \(A D = 3 a\). A particle of mass \(3 m\) is attached to the rod at \(C\), where \(A C = x\). The rod is in equilibrium in a vertical plane perpendicular to the wall as shown in Figure 3. The tension in the string is \(\frac { 25 } { 4 } m g\). Show that

1. \(x = 3 a\),
2. the horizontal component of the force exerted by the wall on the rod has magnitude 5 mg . The coefficient of friction between the wall and the rod is \(\mu\). Given that the rod is about to slip,
3. find the value of \(\mu\).

1. A particle is projected from a point \(O\) with speed \(u\) at an angle of elevation \(\alpha\) above the horizontal and moves freely under gravity. When the particle has moved a horizontal distance \(x\), its height above \(O\) is \(y\).
   1. Show that
   $$y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } \cos ^ { 2 } \alpha }$$ A girl throws a ball from a point \(A\) at the top of a cliff. The point \(A\) is 8 m above a horizontal beach. The ball is projected with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(45 ^ { \circ }\). By modelling the ball as a particle moving freely under gravity,
2. find the horizontal distance of the ball from \(A\) when the ball is 1 m above the beach. A boy is standing on the beach at the point \(B\) vertically below \(A\). He starts to run in a straight line with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), leaving \(B 0.4\) seconds after the ball is thrown. He catches the ball when it is 1 m above the beach.
3. Find the value of \(v\).