A-LevelMathematicsMechanics9709/43 · Oct/Nov 2022 · Q712 Marks

Particles of masses 1.5kg and 3kg lie on a plane which is inclined at an angle of α to the horizontal, where tan α = 3/4. The section of the plane from A to B is smooth and the section of the plane from B to C is rough. The 1.5kg particle is held at rest at A and the 3 kg particle is in limiting equilibrium at B. The distance AB is xm and the distance BC is 4m (see diagram). [Figure X.X]

(a)Show that the coefficient of friction between the particle at B and the plane is 0.75.[3]
(b)The 1.5kg particle is released from rest. In the subsequent motion the two particles collide and coalesce. The time taken for the combined particle to travel from B to C is 2s. The coefficient of friction between the combined particle and the plane is still 0.75. Find x.[6]
(c)Find the total loss of energy of the particles from the time the 1.5 kg particle is released until the combined particle reaches C.[3]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

[B1] R=3gcosa = 3 × 10 × 0.8
[M1] F = 3g sina = 3 × 10 × 0.6 Resolving parallel to plane.
[A1] μ = 18/24 = 0.75 or μ = 3g sin a / 3g cos a = tan a = 0.75 Uses μ = F/R AG.
[B1] a = g sina or PE loss=1.5gx sina for AB and a = 0 for BC 4.5g×sina-0.75×4.5g cosa = 4.5a leading to a = 0 Accelerations for AB and BC.
[M1] v₁² = 2xgsinaxx ] or [1.5g×xsina=0.5×1.5×1₁² Uses 'suvat' or PE loss = KE gain for AB.
[A1] v₁² = 20x sina=12x leading to v₁ = √12x
[M1] 1.5×√12x+0=4.5×v₂ leading to v₂ = 1/3 √12x Conservation of momentum.
[M1] 4= 2/3 x√12x Use of s = vt on BC since a = 0.
[A1] x=3
[B1] KE = 0.5×4.5×2² =9J KE gain for AC.
[M1] PE loss = 15×(4+3)×3+30×4×3=135J Evaluates PE loss for AC.
[A1] Loss of energy = 126 J

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.