Limiting equilibrium on incline

7 \includegraphics[max width=\textwidth, alt={}, center]{ac4bb5a0-c7c0-4e1d-9e76-64f92ae28066-10_214_1461_255_342} As shown in the diagram, particles \(A\) and \(B\) of masses 2 kg and 3 kg respectively are attached to the ends of a light inextensible string. The string passes over a small fixed smooth pulley which is attached to the top of two inclined planes. Particle \(A\) is on plane \(P\), which is inclined at an angle of \(10 ^ { \circ }\) to the horizontal. Particle \(B\) is on plane \(Q\), which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The string is taut, and the two parts of the string are parallel to lines of greatest slope of their respective planes.

1. It is given that plane \(P\) is smooth, plane \(Q\) is rough, and the particles are in limiting equilibrium. Find the coefficient of friction between particle \(B\) and plane \(Q\).
2. It is given instead that both planes are smooth and that the particles are released from rest at the same horizontal level. Find the time taken until the difference in the vertical height of the particles is 1 m . [You should assume that this occurs before \(A\) reaches the pulley or \(B\) reaches the bottom of plane \(Q\).] [6]
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 \includegraphics[max width=\textwidth, alt={}, center]{f0200d12-4ab0-4395-804c-e693f7f26507-2_368_853_1503_644} A small smooth pulley is fixed at the highest point \(A\) of a cross-section \(A B C\) of a triangular prism. Angle \(A B C = 90 ^ { \circ }\) and angle \(B C A = 30 ^ { \circ }\). The prism is fixed with the face containing \(B C\) in contact with a horizontal surface. Particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string, which passes over the pulley. The particles are in equilibrium with \(P\) hanging vertically below the pulley and \(Q\) in contact with \(A C\). The resultant force exerted on the pulley by the string is \(3 \sqrt { } 3 \mathrm {~N}\) (see diagram).

1. Show that the tension in the string is 3 N . The coefficient of friction between \(Q\) and the prism is 0.75 .
2. Given that \(Q\) is in limiting equilibrium and on the point of moving upwards, find its mass.

4 \includegraphics[max width=\textwidth, alt={}, center]{2a91fb7a-0eaf-4c50-8a2c-4755c0b44c17-2_499_784_1617_685} Blocks \(P\) and \(Q\), of mass \(m \mathrm {~kg}\) and 5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane inclined at \(35 ^ { \circ }\) to the horizontal. Block \(P\) is at rest on the plane and block \(Q\) hangs vertically below the pulley (see diagram). The coefficient of friction between block \(P\) and the plane is 0.2 . Find the set of values of \(m\) for which the two blocks remain at rest.

6 Two particles, \(A\) and \(B\), are connected by a light inextensible string which passes over a smooth peg. Particle \(A\) has mass 2 kg and particle \(B\) has mass 4 kg . Particle \(A\) hangs freely with the string vertical. Particle \(B\) is at rest in equilibrium on a rough horizontal surface with the string at an angle of \(30 ^ { \circ }\) to the vertical. The particles, peg and string are shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-14_419_953_571_541}

1. By considering particle \(A\), find the tension in the string.
2. Draw a diagram to show the forces acting on particle \(B\).
3. Show that the magnitude of the normal reaction force acting on particle \(B\) is 22.2 newtons, correct to three significant figures.
4. Find the least possible value of the coefficient of friction between particle \(B\) and the surface.
   \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-16_2486_1714_221_153}