Edexcel M2 (Mechanics 2) 2003 January

1. Three particles of mass \(3 m , 5 m\) and \(\lambda m\) are placed at points with coordinates (4, 0), (0, -3) and \(( 4,2 )\) respectively. The centre of mass of the system of three particles is at \(( 2 , k )\).
   1. Show that \(\lambda = 2\).
   2. Calculate the value of \(k\).
   3. A car of mass 1000 kg is moving along a straight horizontal road with a constant acceleration of \(f \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The resistance to motion is modelled as a constant force of magnitude 1200 N . When the car is travelling at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the power generated by the engine of the car is 24 kW .
   4. Calculate the value of \(f\).
   When the car is travelling at \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the engine is switched off and the car comes to rest, without braking, in a distance of \(d\) metres. Assuming the same model for resistance,
2. use the work-energy principle to calculate the value of \(d\).
3. Give a reason why the model used for the resistance to motion may not be realistic. \section*{3.} \section*{Figure 1}
   \includegraphics[max width=\textwidth, alt={}]{19f831ad-5e32-470c-9974-beb82d5c9753-3_751_678_440_657}
   A uniform ladder \(A B\), of mass \(m\) and length \(2 a\), has one end \(A\) on rough horizontal ground. The other end \(B\) rests against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall. The ladder makes an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac { 4 } { 3 }\). A child of mass \(2 m\) stands on the ladder at \(C\) where \(A C = \frac { 1 } { 2 } a\), as shown in Fig. 1. The ladder and the child are in equilibrium. By modelling the ladder as a rod and the child as a particle, calculate the least possible value of the coefficient of friction between the ladder and the ground.

4. Figure 2
\includegraphics[max width=\textwidth, alt={}, center]{19f831ad-5e32-470c-9974-beb82d5c9753-4_574_1159_395_429} Figure 2 shows a uniform lamina \(A B C D E\) such that \(A B D E\) is a rectangle, \(B C = C D , A B = 8 a\) and \(A E = 6 a\). The point \(X\) is the mid-point of \(B D\) and \(X C = 4 a\). The centre of mass of the lamina is at \(G\).

1. Show that \(G X = \frac { 44 } { 15 } a\).
   (6) The mass of the lamina is \(M\). A particle of mass \(\lambda M\) is attached to the lamina at \(C\). The lamina is suspended from \(B\) and hangs freely under gravity with \(A B\) horizontal.
2. Find the value of \(\lambda\).
   (3)

5. A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\) at time \(t\) seconds is \(( 4 t - 8 ) \mathrm { m } \mathrm { s } ^ { - 2 }\), measured in the direction of \(x\) increasing. The velocity of \(P\) at time \(t\) seconds is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that \(v = 6\) when \(t = 0\), find

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

6. A smooth sphere \(P\) of mass \(2 m\) is moving in a straight line with speed \(u\) on a smooth horizontal table. Another smooth sphere \(Q\) of mass \(m\) is at rest on the table. The sphere \(P\) collides directly with \(Q\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 3 }\). The spheres are modelled as particles.

1. Show that, immediately after the collision, the speeds of \(P\) and \(Q\) are \(\frac { 5 } { 9 } u\) and \(\frac { 8 } { 9 } u\) respectively. After the collision, \(Q\) strikes a fixed vertical wall which is perpendicular to the direction of motion of \(P\) and \(Q\). The coefficient of restitution between \(Q\) and the wall is \(e\). When \(P\) and \(Q\) collide again, \(P\) is brought to rest.
2. Find the value of \(e\).
3. Explain why there must be a third collision between \(P\) and \(Q\).

7. \begin{figure}[h] \end{figure} A ball \(B\) of mass 0.4 kg is struck by a bat at a point \(O\) which is 1.2 m above horizontal ground. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are respectively horizontal and vertical. Immediately before being struck, \(B\) has velocity \(( - 20 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after being struck it has velocity \(( 15 \mathbf { i } + 16 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). After \(B\) has been struck, it moves freely under gravity and strikes the ground at the point \(A\), as shown in Fig. 3. The ball is modelled as a particle.

1. Calculate the magnitude of the impulse exerted by the bat on \(B\).
2. By using the principle of conservation of energy, or otherwise, find the speed of \(B\) when it reaches \(A\).
3. Calculate the angle which the velocity of \(B\) makes with the ground when \(B\) reaches \(A\).
4. State two additional physical factors which could be taken into account in a refinement of the model of the situation which would make it more realistic.