Edexcel M2 (Mechanics 2) 2007 January

1. A particle of mass 0.8 kg is moving in a straight line on a rough horizontal plane. The speed of the particle is reduced from \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as the particle moves 20 m . Assuming that the only resistance to motion is the friction between the particle and the plane, find
   1. the work done by friction in reducing the speed of the particle from \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
   2. the coefficient of friction between the particle and the plane.
      
   3. A car of mass 800 kg is moving at a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 24 }\). The resistance to motion from non-gravitational forces is modelled as a constant force of magnitude 900 N .
   4. Find, in kW , the rate of working of the engine of the car.
   When the car is travelling down the road at \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the engine is switched off. The car comes to rest in time \(T\) seconds after the engine is switched off. The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude 900 N .
2. Find the value of \(T\).

3. \begin{figure}[h] \end{figure} Figure 1 shows a template \(T\) made by removing a circular disc, of centre \(X\) and radius 8 cm , from a uniform circular lamina, of centre \(O\) and radius 24 cm . The point \(X\) lies on the diameter \(A O B\) of the lamina and \(A X = 16 \mathrm {~cm}\). The centre of mass of \(T\) is at the point \(G\).

1. Find \(A G\). The template \(T\) is free to rotate about a smooth fixed horizontal axis, perpendicular to the plane of \(T\), which passes through the mid-point of \(O B\). A small stud of mass \(\frac { 1 } { 4 } m\) is fixed at \(B\), and \(T\) and the stud are in equilibrium with \(A B\) horizontal. Modelling the stud as a particle,
2. find the mass of \(T\) in terms of \(m\).

4. A particle \(P\) of mass \(m\) is moving in a straight line on a smooth horizontal table. Another particle \(Q\) of mass \(k m\) is at rest on the table. The particle \(P\) collides directly with \(Q\). The direction of motion of \(P\) is reversed by the collision. After the collision, the speed of \(P\) is \(v\) and the speed of \(Q\) is \(3 v\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 2 }\).

1. Find, in terms of \(v\) only, the speed of \(P\) before the collision.
2. Find the value of \(k\). After being struck by \(P\), the particle \(Q\) collides directly with a particle \(R\) of mass \(11 m\) which is at rest on the table. After this second collision, \(Q\) and \(R\) have the same speed and are moving in opposite directions. Show that
3. the coefficient of restitution between \(Q\) and \(R\) is \(\frac { 3 } { 4 }\),
4. there will be a further collision between \(P\) and \(Q\).

5. \begin{figure}[h] \end{figure} A horizontal uniform \(\operatorname { rod } A B\) has mass \(m\) and length \(4 a\). The end \(A\) rests against a rough vertical wall. A particle of mass \(2 m\) is attached to the rod at the point \(C\), where \(A C = 3 a\). One end of a light inextensible string \(B D\) is attached to the rod at \(B\) and the other end is attached to the wall at a point \(D\), where \(D\) is vertically above \(A\). The rod is in equilibrium in a vertical plane perpendicular to the wall. The string is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 3 } { 4 }\), as shown in Figure 2.

1. Find the tension in the string.
2. Show that the horizontal component of the force exerted by the wall on the rod has magnitude \(\frac { 8 } { 3 } m g\). The coefficient of friction between the wall and the rod is \(\mu\). Given that the rod is in limiting equilibrium,
3. find the value of \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{7ae16b00-d388-4c1b-a195-c785a3900548-08_158_136_2595_1822}

6. A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, \(\mathbf { F } = \left( 1.5 t ^ { 2 } - 3 \right) \mathbf { i } + 2 t \mathbf { j }\). When \(t = 2\), the velocity of \(P\) is \(( - 4 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the acceleration of \(P\) at time \(t\) seconds.
2. Show that, when \(t = 3\), the velocity of \(P\) is \(( 9 \mathbf { i } + 15 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 3\), the particle \(P\) receives an impulse \(\mathbf { Q }\) Ns. Immediately after the impulse the velocity of \(P\) is \(( - 3 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
3. the magnitude of \(\mathbf { Q }\),
4. the angle between \(\mathbf { Q }\) and \(\mathbf { i }\).

7. \begin{figure}[h] \end{figure} A particle \(P\) is projected from a point \(A\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta\), where \(\cos \theta = \frac { 4 } { 5 }\). The point \(B\), on horizontal ground, is vertically below \(A\) and \(A B = 45 \mathrm {~m}\). After projection, \(P\) moves freely under gravity passing through a point \(C , 30 \mathrm {~m}\) above the ground, before striking the ground at the point \(D\), as shown in Figure 3. Given that \(P\) passes through \(C\) with speed \(24.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),

1. using conservation of energy, or otherwise, show that \(u = 17.5\),
2. find the size of the angle which the velocity of \(P\) makes with the horizontal as \(P\) passes through \(C\),
3. find the distance \(B D\).