A-LevelMathematicsMechanics9709/41 · Oct/Nov 2025 · Q56 Marks

A particle P of mass mkg is in equilibrium on a rough plane inclined at an angle θ° to the horizontal. The equilibrium of P is maintained by a force of magnitude 8mgN making an angle 0° with a line of greatest slope (see diagram). [Figure 5.1] The coefficient of friction between P and the plane is 0.5 and P is on the point of slipping down the plane. Find the value of θ.

A particle P of mass mkg is in equilibrium on a rough plane inclined at an angle θ° to the horizontal. The equilibrium of P is maintained by a force of magnitude 8mgN making an angle 0° with a line of greatest slope (see diagram). [Figure 5.1] The coefficient of friction between P and the plane is 0.5 and P is on the point of slipping down the plane. Find the value of θ.[6]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.

[*M1] Attempt at resolving parallel or perpendicular to the plane to form an equation Correct number of terms, allow sign errors, allow sin/cos mix, allow g missing on the weight term only (so not missing on the 8mg term). Forces that need resolving should be resolved, forces that do not need resolving should not be resolved.
[A1] 8mg cos θ + mg sin θ = F Allow with their possibly incorrect F .
[A1] R = mg cos θ + 8mg sin θ
[DM1] Use of F=0.5R to get an equation in θ [m and g] Where R is a linear combination of a weight (or mass) component and an 8mg component.
[DM1] Re-arranging to tanθ=k where k≠ 0 (or equivalent method to reduce to a single trig. term). dependent on both previous M1 marks.
[A1] θ = 68.2

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