Fig. 6.1 shows particles A and B, of masses 4kg and 3kg respectively, attached to the ends of a light inextensible string that passes over a small smooth pulley. The pulley is fixed at the top of a plane which is inclined at an angle of 30° to the horizontal. A hangs freely below the pulley and B is on the inclined plane. The string is taut and the section of the string between B and the pulley is parallel to a line of greatest slope of the plane.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

1. [B1] T = 4g soi
2. [B1] R = 3g cos 30
3. [M1] Attempt to resolve parallel to the plane 3 terms, allow g missing. Allow sign errors, sin/cos mix.
4. [*A1] F = T - 3g sin 30 May see F = 25.
5. [DM1] Eliminate T and use F = μR to get an equation in μ only Where R is a component of their weight.
6. [A1] Coefficient of friction = 0.962 allow 5√3/9 allow 0.96. If F negative must say why using positive for this mark.
7. [M1] Find height gained by B relative to height lost by A A loses x m in height, B gains x sin 30 OR B gains y m in height and A loses y/sin 30
8. [A1] EITHER x+xsin30=1⇒x=2/3 OR y+y/sin30=1⇒y=1/3
9. [B1] Change in KE = (1/2) * 4v² + (1/2) * 3v² = (1/2) * 7v²
10. [B1] Change in PE = ±(4gx - 3gx sin 30) or ± (4g(y/sin30) - 3gy) OR ±(4gx - 3gy)
11. [M1] Conservation of energy 4gx - 3gx sin 30 = (1/2) * 4v² + (1/2) * 3v² 4 terms. x or y need not be substituted. Must be same v for both particles.
12. [A1] Speed = √(100/21) = 10√21 / 21 = 2.18 m s⁻¹ 2.182178902 SC B1 B1 M1 3/6 max for using x = y = 0.5