A particle of mass 0.6kg is placed on a rough plane which is inclined at an angle of 35° to the horizontal. The particle is kept in equilibrium by a horizontal force of magnitude PN acting in a vertical plane containing a line of greatest slope (see diagram). The coefficient of friction between the particle and plane is 0.4. Find the least possible value of P. [Figure 5.1]

Mark scheme

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

1. Attempt at resolving parallel or perpendicular to the plane.
2. R = Psin 35 +0.6g cos35 [R = (0.573...) P+4.914...]
3. [A1]
4. F + Pcos35=0.6g sin35 [F+(0.819...)P=3.441...]
5. [A1] Their F.
6. Use of F = 0.4R
7. [M1] Where R is initially a linear combination of a P component and a weight component (or a mass component).
8. Solve for P.
9. [DM1] From equations with the correct number of relevant resolved terms. May see R = 0.6g / (cos 35+0.4 sin 35) = 5.7222. Must get to P = ..., e.g. P = (0.6g sin 35 - 0.4×0.6g cos 35) / (cos 35 + 0.4 sin 35). If no working seen, allow this mark if correct solution for their equations. If F < 0.4R used, it should be used correctly. e.g. 0.6g sin 35 – P cos 35 ≤ 0.4(P sin 35 + 0.6g cos 35).
10. P=1.41
11. [A1] AWRT 1.41. If P ≥ 1.41 seen, must then state the least value explicitly for A1.
12. Alternative for Question 5: Resolving vertically and horizontally
13. Attempt at resolving vertically or horizontally.
14. Rcos 35+ F sin 35 = 0.6g
15. [A1] Their F or R.
16. P+F cos 35 = R sin 35
17. [A1] Their F or R.
18. Use of F = 0.4R
19. [M1] To get 2 equations, one in R (or F) and the other in P and R (or P and F) from resolved equations with correct number of relevant terms. Allow g missing.
20. Solve for P
21. [DM1] From equations with the correct number of relevant resolved terms. May see R = 0.6g / (cos 35+0.4 sin 35) = 5.7222; Must get to P = ..., e.g. P = (0.6g sin 35 - 0.4×0.6g cos 35) / (cos 35 + 0.4 sin 35). If no working seen, allow this mark if correct solution for their equations.
22. P=1.41
23. [A1] AWRT 1.41.