A-LevelMathematicsMechanics9709/42 · Feb/Mar 2025 · Q610 Marks

A block of mass 12kg is placed on a rough plane inclined at an angle of α to the horizontal, where α = tan⁻¹0.5. A force of X N is applied to the block, directly up the plane (see diagram). The coefficient of friction between the block and the plane is μ. [Figure X.X]

(a)It is given that μ = 0.15 and X = 20. Find the time that it takes for the block to move 2m down the plane from rest.[6]
(b)It is given instead that μ ≠ 0.15 and that when X = 10, the block is on the point of moving down the plane. Find the value of μ and the value of X for which the block is on the point of moving up the plane.[4]

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Mark scheme

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

B1 Allow cos27 or better for 2/√5. α = 26.56505118.
*M1 For use of N2L with 4 terms. Allow sign errors. Allow sin/cos mix. Allow g missing Allow sin 27 or better for 1/√5. Allow their possibly incorrect F.
DM1 For use of F=0.15R to get an equation in a only, where R is a component of weight or mass.
A1 SOI. Allow AWRT 1.5 a=1.46382...
DM1 Dependent on both M marks. For use of constant acceleration to find t. Allow their a.
A1 t=1.65304... Allow 1.66 from using a=1.46.
*M1 3 terms; allow sin/cos mix.
A1 F = 43.7[43.665...]. Allow sin27 or better for 1/√5. Allow cos 27 or better for 2/√5.
DM1 Solving for μ must be using R as a component of weight. From equation(s) with the correct number of relevant terms and no sign errors.
A1 X=97.3 and μ = 0.407. Allow X=97.4 or 97.5 from correct work. Allow μ = 0.408 from correct work.

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