Challenging +1.2 This is a multi-step friction problem requiring resolution of forces in two directions, consideration of limiting friction inequality (F ≤ μR), and solving simultaneous inequalities to find a range for P. While it involves several components (inclined plane, angled force, friction limits), the techniques are standard M1 material with no novel insight required—just careful systematic application of equilibrium conditions and friction laws.
\includegraphics{figure_5}
A particle of mass \(0.12\) kg is placed on a plane which is inclined at an angle of \(40°\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(P\) N acting up the plane at an angle of \(30°\) above a line of greatest slope, as shown in the diagram. The coefficient of friction between the particle and the plane is \(0.32\). Find the set of possible values of \(P\).
[8]
Substitute for F and solve for P in either case, 4 terms
OR solve a pair of simultaneous equations (each with 3
terms) in R and P for P in one of the cases
P = 1.04 P = 0.676
Answer
Marks
Guidance
max min
A1
For either correct
0.676 ⩽ P ⩽ 1.04
A1
Total:
8
Question
Answer
Marks
Question 5:
5 | M1 | Resolve perpendicular to the plane, three terms
R + P sin 30 = 0.12g cos 40 | A1 | R does not need to be the subject
F = 0.32R | M1 | Use F = µR
[P cos 30 + F = 0.12g sin 40]
min | M1 | About to slip down, 3 terms
[P cos 30 – F = 0.12g sin 40]
max | M1 | About to slip up, 3 terms
[P cos 30 = 0.12g sin 40
±0.32 (0.12g cos 40 – P sin 30)]
OR
[Pcos 30 ± 0.32R = 0.12g sin 40
R + P sin 30 = 0.12g cos 40]
Must reach P =… in either method | M1 | Substitute for F and solve for P in either case, 4 terms
OR solve a pair of simultaneous equations (each with 3
terms) in R and P for P in one of the cases
P = 1.04 P = 0.676
max min | A1 | For either correct
0.676 ⩽ P ⩽ 1.04 | A1
Total: | 8
Question | Answer | Marks | Guidance
\includegraphics{figure_5}
A particle of mass $0.12$ kg is placed on a plane which is inclined at an angle of $40°$ to the horizontal. The particle is kept in equilibrium by a force of magnitude $P$ N acting up the plane at an angle of $30°$ above a line of greatest slope, as shown in the diagram. The coefficient of friction between the particle and the plane is $0.32$. Find the set of possible values of $P$.
[8]
\hfill \mbox{\textit{CAIE M1 2017 Q5 [8]}}