Particles A and B, of masses 2.4 kg and 3.3 kg respectively, are connected by a light inextensible string that passes over a smooth pulley which is fixed to the top of a rough plane. The plane makes an angle of θ° with horizontal ground. Particle A is on the plane and the section of the string between A and the pulley is parallel to a line of greatest slope of the plane. Particle B hangs vertically below the pulley and is 1 m above the ground [Figure 7.1]. The coefficient of friction between the plane and A is μ.

Mark scheme

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

1. [M1*] Resolving for both particles or for the system to form equation(s) Must have correct number of terms. Allow sign errors. Allow sin/cos mix. Allow g missing. M0 if acceleration included unless subsequently equated to zero. Masses must be appropriate for their equation(s). Forces must have components (or not) as required.
2. [A1] Either T-F-2.4g sin30=0 AND 3.3g-T = 0 Or 3.3g-F-2.4g sin30=0 Both correct or system correct. May get F = 21. Can be with a wrong non-zero F.
3. [B1] R=2.4gcos30 [=12√3=20.7846...]
4. [DM1] Use of F = µR to get an equation in μ only. Must be from F dimensionally correct and single term R which is equal to a component the 2.4 kg weight. Allow consistent sin/cos mix but must be different components of weight. F and R must be numerical expressions. (e.g., [3.3g - 2.4gpcos 30 -2.4g sin 30 = 0])
5. [A1] μ=1.01 [sight of 1.01036... or 1.0104] AG perhaps from one of (3.3g-24sin30)/(2.4g cos 30) = (33-12)/(12√3) = 21/(12√3) = 7√3/12 = 21/20.7846... = 21/20.8. Do not allow unless evidence of 30 substituted for θ. E.g.: sight of 1.01036... or 1.0104.
6. [M1*] [For 7(b)] Using Newton's second law for both particles or the system Must have correct number of terms. Allow sign errors. Allow sin/cos mix. Allow g missing. Masses must be appropriate for their equation(s). Forces must have components (or not) as required.
7. [A1] Either 3.3g-T=3.3a and T - F -2.4g sin 20 = 2.4a Or 3.3g - F -2.4g sin20=(2.4+3.3)a Both correct or system equation correct. Can be with a wrong non-zero F.
8. [B1] F = 1.01×2.4g cos 20 [= 22.778] For correct expression for F.
9. [DM1] Attempt to solve for a a=0.353 [0.353222...] Using their F Must get to ‘a ='. If sin/cos mix must be consistent.
10. [A1FT] v² = 2×0.353×1 [= 0.706444...] or [v = 0.841] FT their value of a≠±g to get an expression for v² or v. Can be implied by awrt 0.84 for v or awrt 0.71 for v² . This mark does not depend on previous A or B mark, but both Ms must have been awarded. Or 1=0+(1/2)×0.353t²⇒t=2.3795⇒v=0.353×2.38
11. [M1*] Using Newton's second law on A after B reaches the ground -F-2.4g sin 20 = 2.4a Must have correct number of terms. Allow sign errors. Allow sin/cos mix. Allow g missing. [⇒a=-12.911...]
12. [DM1] Use of suvat to find s [0=their0.841² +2×their-12.911...×s ⇒s=0.027358....] Using their a≠±g. Must get to 's ='. May find and use t = 0.0651.
13. [A1] Total distance = 1.03 m
14. [B1] [Alternative method using energy for 7(b), first 5 marks] [KE gained =](1/2)×(2.4+3.3)v² [= 2.85v²]
15. [B1] [PE lost =]3.3g×1-2.4g×1sin20[= 24.791...] Allow omission of 1 in either or both terms.
16. [B1] [Friction =] 1.01×2.4g cos 20[= 22.778...] For correct expression for F.
17. [M1] Attempt for energy equation. Allow sign errors, allow sin/cos mix but must have sin/cos where needed. Correct number of terms, dimensionally correct. Allow omission of 1 in any or all the three relevant terms. Must have cos 20 and sin 20. (e.g., (1/2)×(2.4+3.3)v² = 3.3g×1-2.4g×1sin20-1.01×2.4gcos20×1 or 2.85v² = 24.791...-22.778...)
18. [A1] To get a correct expression for v² [v² = 0.706444... or v = 0.841...] Can be implied by awrt 0.84 for v or awrt 0.71 for v² if expression not seen.
19. [M1] [Alternative method using energy for 7(b), final 3 marks] KE = (1/2)x2.4×0.841² Using their v².
20. [M1] 1.01×2.4gcos20×s + 2.4gsin20×s = (1/2)×2.4×0.841² [⇒ s = 0.027358..] For attempt at 3 term energy equation and solved to get to 's ='. Allow sign errors, allow consistent sin/cos mix but must have sin/cos where needed. Correct number of terms, dimensionally correct.
21. [A1] Total distance = 1.03 m