1 A particle of mass 0.4 kg is projected with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal.

1. Find the initial kinetic energy of the particle.
2. Use an energy method to find the distance the particle moves up the plane before coming to instantaneous rest.
   \includegraphics[max width=\textwidth, alt={}, center]{f3ddfde8-b678-4c3a-a9a6-d984c2c38bd9-03_266_874_260_632} A particle \(P\) of mass 1.6 kg is suspended in equilibrium by two light inextensible strings attached to points \(A\) and \(B\). The strings make angles of \(20 ^ { \circ }\) and \(40 ^ { \circ }\) respectively with the horizontal (see diagram). Find the tensions in the two strings.
   \includegraphics[max width=\textwidth, alt={}, center]{f3ddfde8-b678-4c3a-a9a6-d984c2c38bd9-04_286_664_251_737} A particle of mass 0.6 kg is placed on a rough plane which is inclined at an angle of \(21 ^ { \circ }\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(P \mathrm {~N}\) acting parallel to a line of greatest slope of the plane, as shown in the diagram. The coefficient of friction between the particle and the plane is 0.3 . Show that the least possible value of \(P\) is 0.470 , correct to 3 significant figures, and find the greatest possible value of \(P\).