3. A particle \(P\) of mass 0.3 kg is moving under the action of a single force F newtons. At time \(t\) seconds the velocity of \(P , \mathbf { v } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$\mathbf { v } = 3 t ^ { 2 } \mathbf { i } + ( 6 t - 4 ) \mathbf { j } .$$

1. Calculate, to 3 significant figures, the magnitude of \(\mathbf { F }\) when \(t = 2\). When \(t = 0 , P\) is at the point \(A\). The position vector of \(A\) with respect to a fixed origin \(O\) is \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m }\). When \(t = 4 , P\) is at the point \(B\).
2. Find the position vector of \(B\). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d5250c24-138a-4fe3-91eb-d6f8dca766fb-3_993_1099_391_468}
   \end{figure} Figure 1 shows a template made by removing a square \(W X Y Z\) from a uniform triangular lamina \(A B C\). The lamina is isosceles with \(C A = C B\) and \(A B = 12 a\). The mid-point of \(A B\) is \(N\) and \(N C = 8 a\). The centre \(O\) of the square lies on \(N C\) and \(O N = 2 a\). The sides \(W X\) and \(Z Y\) are parallel to \(A B\) and \(W Z = 2 a\). The centre of mass of the template is at \(G\).
3. Show that \(N G = \frac { 30 } { 11 } a\). The template has mass \(M\). A small metal stud of mass \(k M\) is attached to the template at \(C\). The centre of mass of the combined template and stud lies on \(Y Z\). By modelling the stud as a particle,
4. calculate the value of \(k\). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d5250c24-138a-4fe3-91eb-d6f8dca766fb-4_873_1125_352_519}
   \end{figure} Figure 2 shows a horizontal uniform pole \(A B\), of weight \(W\) and length \(2 a\). The end \(A\) of the pole rests against a rough vertical wall. One end of a light inextensible string \(B D\) is attached to the pole at \(B\) and the other end is attached to the wall at \(D\). A particle of weight \(2 W\) is attached to the pole at \(C\), where \(B C = x\). The pole is in equilibrium in a vertical plane perpendicular to the wall. The string \(B D\) is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\). The pole is modelled as a uniform rod.
5. Show that the tension in \(B D\) is \(\frac { 5 ( 5 a - 2 x ) } { 6 a } W\). The vertical component of the force exerted by the wall on the pole is \(\frac { 7 } { 6 } W\). Find (b) \(x\) in terms of \(a\),
6. the horizontal component, in terms of \(W\), of the force exerted by the wall on the pole.