Edexcel M2 (Mechanics 2) 2011 January

1. A cyclist starts from rest and moves along a straight horizontal road. The combined mass of the cyclist and his cycle is 120 kg . The resistance to motion is modelled as a constant force of magnitude 32 N . The rate at which the cyclist works is 384 W . The cyclist accelerates until he reaches a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

Find

1. the value of \(v\),
2. the acceleration of the cyclist at the instant when the speed is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2. A particle of mass 2 kg is moving with velocity \(( 5 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse of \(( - 6 \mathbf { i } + 8 \mathbf { j } ) \mathrm { N }\) s. Find the kinetic energy of the particle immediately after receiving the impulse.
(5)
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3. A particle moves along the \(x\)-axis. At time \(t = 0\) the particle passes through the origin with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. The acceleration of the particle at time \(t\) seconds, \(t \geqslant 0\), is \(\left( 4 t ^ { 3 } - 12 t \right) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the positive \(x\)-direction. Find

1. the velocity of the particle at time \(t\) seconds,
2. the displacement of the particle from the origin at time \(t\) seconds,
3. the values of \(t\) at which the particle is instantaneously at rest.

4. \begin{figure}[h] \end{figure} A box of mass 30 kg is held at rest at point \(A\) on a rough inclined plane. The plane is inclined at \(20 ^ { \circ }\) to the horizontal. Point \(B\) is 50 m from \(A\) up a line of greatest slope of the plane, as shown in Figure 1. The box is dragged from \(A\) to \(B\) by a force acting parallel to \(A B\) and then held at rest at \(B\). The coefficient of friction between the box and the plane is \(\frac { 1 } { 4 }\). Friction is the only non-gravitational resistive force acting on the box. Modelling the box as a particle,

1. find the work done in dragging the box from \(A\) to \(B\). The box is released from rest at the point \(B\) and slides down the slope. Using the workenergy principle, or otherwise,
2. find the speed of the box as it reaches \(A\).
   January 2011

5. \begin{figure}[h] \end{figure} The uniform L-shaped lamina \(A B C D E F\), shown in Figure 2, has sides \(A B\) and \(F E\) parallel, and sides \(B C\) and \(E D\) parallel. The pairs of parallel sides are 9 cm apart. The points \(A , F\), \(D\) and \(C\) lie on a straight line.
\(A B = B C = 36 \mathrm {~cm} , F E = E D = 18 \mathrm {~cm} . \angle A B C = \angle F E D = 90 ^ { \circ }\), and \(\angle B C D = \angle E D F = \angle E F D = \angle B A C = 45 ^ { \circ }\).

1. Find the distance of the centre of mass of the lamina from
   1. side \(A B\),
   2. side \(B C\). The lamina is freely suspended from \(A\) and hangs in equilibrium.
2. Find, to the nearest degree, the size of the angle between \(A B\) and the vertical.

1. \hspace{0pt} [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.]

\begin{figure}[h] \end{figure} At time \(t = 0\), a particle \(P\) is projected from the point \(A\) which has position vector 10j metres with respect to a fixed origin \(O\) at ground level. The ground is horizontal. The velocity of projection of \(P\) is \(( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), as shown in Figure 3. The particle moves freely under gravity and reaches the ground after \(T\) seconds.

1. For \(0 \leqslant t \leqslant T\), show that, with respect to \(O\), the position vector, \(\mathbf { r }\) metres, of \(P\) at time \(t\) seconds is given by $$\mathbf { r } = 3 t \mathbf { i } + \left( 10 + 5 t - 4.9 t ^ { 2 } \right) \mathbf { j }$$
2. Find the value of \(T\).
3. Find the velocity of \(P\) at time \(t\) seconds \(( 0 \leqslant t \leqslant T )\). When \(P\) is at the point \(B\), the direction of motion of \(P\) is \(45 ^ { \circ }\) below the horizontal.
4. Find the time taken for \(P\) to move from \(A\) to \(B\).
5. Find the speed of \(P\) as it passes through \(B\).

7. \begin{figure}[h] \end{figure} A uniform plank \(A B\), of weight 100 N and length 4 m , rests in equilibrium with the end \(A\) on rough horizontal ground. The plank rests on a smooth cylindrical drum. The drum is fixed to the ground and cannot move. The point of contact between the plank and the drum is \(C\), where \(A C = 3 \mathrm {~m}\), as shown in Figure 4. The plank is resting in a vertical plane which is perpendicular to the axis of the drum, at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 3 }\). The coefficient of friction between the plank and the ground is \(\mu\). Modelling the plank as a rod, find the least possible value of \(\mu\).

1. A particle \(P\) of mass \(m \mathrm {~kg}\) is moving with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on a smooth horizontal floor. The particle strikes a fixed smooth vertical wall at right angles and rebounds. The kinetic energy lost in the impact is 64 J . The coefficient of restitution between \(P\) and the wall is \(\frac { 1 } { 3 }\).
   1. Show that \(m = 4\).
      (6)
   After rebounding from the wall, \(P\) collides directly with a particle \(Q\) which is moving towards \(P\) with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The mass of \(Q\) is 2 kg and the coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 3 }\).
2. Show that there will be a second collision between \(P\) and the wall.