Two particles P and Q, of masses 0.2kg and 0.1 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley at B which is attached to two inclined planes. Particle P lies on a smooth plane AB which is inclined at 60° to the horizontal. Particle Q lies on a plane BC which is inclined at an angle of θ° to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes (see diagram). [Figure 6.1]

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

1. [B1] Correctly resolving perpendicular to the plane for Q.
2. [M1] Use of F = µR for Q where R is a component of weight but not mass; allow sin/cos mix.
3. [M1] Complete method to determine the resultant force for the whole system. Allow sign errors and sin/cos mix, but must include all required terms and be dimensionally correct. If considering either the while system or P and Q separately then ignore the RHS of their Newton's second law equations.
4. [A1] Correct indication (with no incorrect working) that the resultant force is positive (e.g. 0.8660... -0.35 > 0 or 0.516... (to at least 1 sf) which is positive) together with a correct conclusion. Candidates may calculate the acceleration which is (10√3-7)/6 ≈ 1.72008... and then say that the particles are moving.
5. [M1] Attempt to use Newton's second law for P: 0.2g sin 60-(√3-1) = 0.2a Allow sign errors, sin/cos mix. but must be dimensionally consistent.
6. [A1]
7. [M1] Attempt Newton's second law for Q, or for the whole system. Allow sign errors, sin/cos mix, but must be dimensionally consistent.
8. [A1] 13.41784...