Edexcel M1 (Mechanics 1) 2017 June

1. Three forces, \(( 15 \mathbf { i } + \mathbf { j } ) \mathrm { N } , ( 5 q \mathbf { i } - p \mathbf { j } ) \mathrm { N }\) and \(( - 3 p \mathbf { i } - q \mathbf { j } ) \mathrm { N }\), where \(p\) and \(q\) are constants, act on a particle. Given that the particle is in equilibrium, find the value of \(p\) and the value of \(q\).
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2. Two particles, \(P\) and \(Q\), have masses \(2 m\) and \(3 m\) respectively. They are moving towards each other in opposite directions on a smooth horizontal plane when they collide directly. Immediately before they collide the speed of \(P\) is \(4 u\) and the speed of \(Q\) is \(3 u\). As a result of the collision, \(Q\) has its direction of motion reversed and is moving with speed \(u\).

1. Find the speed of \(P\) immediately after the collision.
2. State whether or not the direction of motion of \(P\) has been reversed by the collision.
3. Find the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision.

3. A plank \(A B\) has length 6 m and mass 30 kg . The point \(C\) is on the plank with \(C B = 2 \mathrm {~m}\). The plank rests in equilibrium in a horizontal position on supports at \(A\) and \(C\). Two people, each of mass 75 kg , stand on the plank. One person stands at the point \(P\) of the plank, where \(A P = x\) metres, and the other person stands at the point \(Q\) of the plank, where \(A Q = 2 x\) metres. The plank remains horizontal and in equilibrium with the magnitude of the reaction at \(C\) five times the magnitude of the reaction at \(A\). The plank is modelled as a uniform rod and each person is modelled as a particle.

1. Find the value of \(x\).
2. State two ways in which you have used the assumptions made in modelling the plank as a uniform rod.

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass 5 kg is held at rest in equilibrium on a rough inclined plane by a horizontal force of magnitude 10 N . The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Figure 1. The line of action of the force lies in the vertical plane containing \(P\) and a line of greatest slope of the plane. The coefficient of friction between \(P\) and the plane is \(\mu\). Given that \(P\) is on the point of sliding down the plane, find the value of \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{c809d34e-83db-4a16-a831-001f9f36b1c3-13_2460_72_311_27}

5. \begin{figure}[h] \end{figure} A vertical light rod \(P Q\) has a particle of mass 0.5 kg attached to it at \(P\) and a particle of mass 0.75 kg attached to it at \(Q\), to form a system, as shown in Figure 2. The system is accelerated vertically upwards by a vertical force of magnitude 15 N applied to the particle at \(Q\). Find the thrust in the rod.

6. A cyclist is moving along a straight horizontal road and passes a point \(A\). Five seconds later, at the instant when she is moving with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), she passes the point \(B\). She moves with constant acceleration from \(A\) to \(B\). Given that \(A B = 40 \mathrm {~m}\), find

1. the acceleration of the cyclist as she moves from \(A\) to \(B\),
2. the time it takes her to travel from \(A\) to the midpoint of \(A B\).

7. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin \(O\).] Two ships, \(P\) and \(Q\), are moving with constant velocities.
The velocity of \(P\) is \(( 9 \mathbf { i } - 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and the velocity of \(Q\) is \(( 4 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\)

1. Find the direction of motion of \(P\), giving your answer as a bearing to the nearest degree. When \(t = 0\), the position vector of \(P\) is \(( 9 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\) and the position vector of \(Q\) is \(( \mathbf { i } + 4 \mathbf { j } ) \mathrm { km }\). At time \(t\) hours, the position vectors of \(P\) and \(Q\) are \(\mathbf { p } \mathrm { km }\) and \(\mathbf { q } \mathrm { km }\) respectively.
2. Find an expression for
   1. \(\mathbf { p }\) in terms of \(t\),
   2. \(\mathbf { q }\) in terms of \(t\).
3. Hence show that, at time \(t\) hours, $$\overrightarrow { Q P } = ( 8 + 5 t ) \mathbf { i } + ( 6 - 10 t ) \mathbf { j }$$
4. Find the values of \(t\) when the ships are 10 km apart.

8. \begin{figure}[h] \end{figure} Two particles, \(A\) and \(B\), have masses \(2 m\) and \(m\) respectively. The particles are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a fixed rough horizontal table at a distance \(d\) from a small smooth light pulley which is fixed at the edge of the table at the point \(P\). The coefficient of friction between \(A\) and the table is \(\mu\), where \(\mu < \frac { 1 } { 2 }\). The string is parallel to the table from \(A\) to \(P\) and passes over the pulley. Particle \(B\) hangs freely at rest vertically below \(P\) with the string taut and at a height \(h\), ( \(h < d\) ), above a horizontal floor, as shown in Figure 3. Particle \(A\) is released from rest with the string taut and slides along the table.

1. 1. Write down an equation of motion for \(A\).
   2. Write down an equation of motion for \(B\).
2. Hence show that, until \(B\) hits the floor, the acceleration of \(A\) is \(\frac { g } { 3 } ( 1 - 2 \mu )\).
3. Find, in terms of \(g , h\) and \(\mu\), the speed of \(A\) at the instant when \(B\) hits the floor. After \(B\) hits the floor, \(A\) continues to slide along the table. Given that \(\mu = \frac { 1 } { 3 }\) and that \(A\) comes to rest at \(P\),
4. find \(d\) in terms of \(h\).
5. Describe what would happen if \(\mu = \frac { 1 } { 2 }\)

   (Total 15 marks)	Leave blank Q8