Two particles P and Q, each of mass m kg, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a rough plane. The plane is inclined at an angle α to the horizontal, where tan α = 24/7. Particle P rests on the plane and particle Q hangs vertically, as shown in the diagram. The string between P and the pulley is parallel to a line of greatest slope of the plane. The system is in limiting equilibrium. [Figure X.X]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

1. [B1] R = mg cos α (R = 9.6m) Allow use of α = 16.3° throughout
2. [M1] [T= mg, F = mg sin α + T] For resolving forces on P and Q and eliminating T or for considering the equilibrium of the system
3. [A1] F = mg sin α + mg (F = 12.8m)
4. [M1] For use of F = μR
5. [A1] Coefficient of friction = ¾ AG so must be from exact working
6. (*M1) P equation is 10 - mg sin α - F - T = 2.5 m or Q equation is T-mg = 2.5m For applying Newton's 2nd law to P (5 terms) or Q (3 terms)
7. (*M1) For applying Newton's 2nd law to the other particle and eliminate T
8. [A1] 10 – mg sin α – μmg cos α – mg = 2m (2.5) If evaluated then this is 10-2.8m-12.8m-10m = 5m
9. [DM1] For solving this equation for m as far as m = Dependent on one or other of the previous M marks having been scored
10. [A1] m = 0.327 Allow m = 50/153