Edexcel M2 (Mechanics 2) 2016 October

1. Three particles of masses \(m , 4 m\) and \(k m\) are placed at the points whose coordinates are \(( - 3,2 ) , ( 4,3 )\) and \(( 6 , - 4 )\) respectively. The centre of mass of the three particles is at the point with coordinates \(( c , 0 )\).

Find

1. the value of \(k\),
2. the value of \(c\).

2. A particle of mass 2 kg is moving with velocity \(3 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives an impulse \(( \lambda \mathbf { i } - 2 \lambda \mathbf { j } )\) Ns, where \(\lambda\) is a constant. Immediately after the impulse is received, the speed of the particle is \(6 \mathrm {~ms} ^ { - 1 }\). Find the possible values of \(\lambda\).

3. A particle \(P\) of mass 4 kg is projected with speed \(6 \mathrm {~ms} ^ { - 1 }\) up a line of greatest slope of a fixed rough inclined plane. The plane is inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 7 }\). The particle is projected from the point \(A\) on the plane and comes to instantaneous rest at the point \(B\) on the plane, where \(A B = 10 \mathrm {~m}\).

1. Show that the work done against friction as \(P\) moves from \(A\) to \(B\) is 16 joules. After coming to instantaneous rest at \(B\), the particle slides back down the plane.
2. Use the work-energy principle to find the speed of \(P\) at the instant it returns to \(A\).

1. At time \(t\) seconds \(( t \geqslant 0 )\), a particle \(P\) has position vector \(\mathbf { r }\) metres with respect to a fixed origin \(O\), where

$$\mathbf { r } = \left( t ^ { 3 } - \frac { 9 } { 2 } t ^ { 2 } - 24 t \right) \mathbf { i } + \left( - t ^ { 3 } + 3 t ^ { 2 } + 12 t \right) \mathbf { j }$$ At time \(T\) seconds, \(P\) is moving in a direction parallel to the vector \(\mathbf { - i } - \mathbf { j }\)
Find

1. the value of \(T\),
2. the magnitude of the acceleration of \(P\) at the instant when \(t = T\).

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\) of length 8 m and weight \(W\) newtons rests in equilibrium against a rough horizontal peg \(P\). The end \(A\) is on rough horizontal ground. The friction is limiting at both \(A\) and \(P\). The distance \(A P\) is 5 m , as shown in Figure 1. The rod rests at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 4 } { 3 }\). The rod is in a vertical plane which is perpendicular to \(P\). The coefficient of friction between the rod and \(P\) is \(\frac { 1 } { 4 }\) and the coefficient of friction between the rod and the ground is \(\mu\).

1. Show that the magnitude of the normal reaction between the rod and \(P\) is \(0.48 W\) newtons.
2. Find the value of \(\mu\).

6. \begin{figure}[h] \end{figure} The uniform lamina \(L\) shown shaded in Figure 2 is formed by removing two circular discs, \(C _ { 1 }\) and \(C _ { 2 }\), from a circular disc with centre \(O\) and radius \(8 a\). Disc \(C _ { 1 }\) has centre \(A\) and radius \(a\). Disc \(C _ { 2 }\) has centre \(B\) and radius \(2 a\). The diameters \(P R\) and \(Q S\) are perpendicular. The midpoint of \(P O\) is \(A\) and the midpoint of \(O R\) is \(B\).

1. Show that the centre of mass of \(L\) is \(\frac { 484 } { 59 } a\) from \(R\). The mass of \(L\) is \(M\). A particle of mass \(k M\) is attached to \(L\) at \(S\). The lamina with the attached particle is suspended from \(R\) and hangs freely in equilibrium with the diameter \(P R\) at an angle of arctan \(\left( \frac { 1 } { 4 } \right)\) to the downward vertical through \(R\).
2. Find the value of \(k\).

7. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards. Position vectors are given relative to a fixed origin O.] At time \(t = 0\) seconds, the particle \(P\) is projected from \(O\) with velocity ( \(3 \mathbf { i } + \lambda \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\), where \(\lambda\) is a positive constant. The particle moves freely under gravity. As \(P\) passes through the fixed point \(A\) it has velocity \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The kinetic energy of \(P\) at the instant it passes through \(A\) is half the initial kinetic energy of \(P\). Find the position vector of \(A\), giving the components to 2 significant figures.
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8. Particles \(A , B\) and \(C\), of masses \(4 m , k m\) and \(2 m\) respectively, lie at rest in a straight line on a smooth horizontal surface with \(B\) between \(A\) and \(C\). Particle \(A\) is projected towards particle \(B\) with speed \(3 u\) and collides directly with \(B\). The coefficient of restitution between each pair of particles is \(\frac { 2 } { 3 }\) Find

1. the speed of \(A\) immediately after the collision with \(B\), giving your answer in terms of \(u\) and \(k\),
2. the range of values of \(k\) for which \(A\) and \(B\) will both be moving in the same direction immediately after they collide. After the collision between \(A\) and \(B\), particle \(B\) collides directly with \(C\). Given that \(k = 4\),
3. show that there will not be a second collision between \(A\) and \(B\).
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