2
\includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-03_293_638_258_717} A particle of mass 7.5 kg , starting from rest at \(A\), slides down an inclined plane \(A B\). The point \(B\) is 12.5 metres vertically below the level of \(A\), as shown in the diagram.

1. Given that the plane is smooth, use an energy method to find the speed of the particle at \(B\).
2. It is given instead that the plane is rough and the particle reaches \(B\) with a speed of \(8 \mathrm {~ms} ^ { - 1 }\). The plane is 25 m long and the constant frictional force has magnitude \(F \mathrm {~N}\). Find the value of \(F\).
   \includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-04_725_655_251_641} Coplanar forces of magnitudes \(52 \mathrm {~N} , 39 \mathrm {~N}\) and \(P \mathrm {~N}\) act at a point in the directions shown in the diagram. The system is in equilibrium. Find the values of \(P\) and \(\theta\).
   \includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-04_2716_38_109_2012}
   \includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-05_2716_29_107_22}