7
6. [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors.] A particle \(A\) of mass 0.5 kg is at rest on a smooth horizontal plane.
At time \(t = 0\), two forces, \(\mathbf { F } _ { 1 } = ( - 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }\), where \(p\) and \(q\) are constants, are applied to \(A\). Given that \(A\) moves in the direction of the vector \(( \mathbf { i } - 2 \mathbf { j } )\),

1. show that \(2 p + q - 4 = 0\) Given that \(p = 5\)
2. find the speed of \(A\) at time \(t = 4\) seconds.
   7. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2633b149-96db-4b80-96c2-e3e6bfbee174-24_451_851_310_493} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string. Another particle \(Q\), also of mass \(m\), is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at the edge of a rough horizontal table. Particle \(Q\) is held at rest on the table and particle \(P\) hangs vertically below the pulley with the string taut, as shown in Figure 4. The pulley, \(P\) and \(Q\) all lie in the same vertical plane.
   The coefficient of friction between \(Q\) and the table is \(\mu\), where \(\mu < 1\)
   Particle \(Q\) is released from rest.
   The tension in the string before \(Q\) hits the pulley is \(k m g\), where \(k\) is a constant.
3. Find \(k\) in terms of \(\mu\). Given that \(Q\) is initially a distance \(d\) from the pulley,
4. find, in terms of \(d , g\) and \(\mu\), the time taken by \(Q\), after release, to reach the pulley.
5. Describe what would happen if \(\mu \geqslant 1\), giving a reason for your answer.