10 A particle \(P\) of mass \(m \mathrm {~kg}\) is moving on a smooth horizontal surface under the action of two constant horizontal forces \(( - 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) and \(( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }\). The resultant of these two forces is \(\mathbf { R } \mathrm { N }\). It is given that \(\mathbf { R }\) acts in a direction which is parallel to the vector \(- \mathbf { i } + 3 \mathbf { j }\).

1. Show that \(3 a + b = 10\). It is given that \(a = 6\) and that \(P\) moves with an acceleration of magnitude \(5 \sqrt { 10 } \mathrm {~ms} ^ { - 2 }\).
2. Determine the value of \(m\).
   \includegraphics[max width=\textwidth, alt={}, center]{977ffad6-2440-46bf-9f17-0f30817d2ddf-08_515_889_248_242} A uniform rod \(A B\), of weight 20 N and length 2.8 m , rests in equilibrium with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth wall inclined at \(55 ^ { \circ }\) to the horizontal. The rod, which rests in a vertical plane that is perpendicular to the wall, is inclined at \(30 ^ { \circ }\) to the horizontal (see diagram).
3. Show that the magnitude of the force acting on the rod at \(B\) is 9.56 N , correct to \(\mathbf { 3 }\) significant figures.
4. Determine the magnitude of the contact force between the rod and the ground. Give your answer correct to \(\mathbf { 3 }\) significant figures.