A-LevelMathematicsMechanics9709/43 · May/June 2022 · Q610 Marks

Two particles P and Q, of masses 0.3kg and 0.2kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley at B which is attached to two inclined planes. P lies on a smooth plane AB which is inclined at 60° to the horizontal. Q lies on a plane BC which is inclined at 30° to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes (see diagram). [Figure 6.1]

(a)It is given that the plane BC is smooth and that the particles are released from rest. Find the tension in the string and the magnitude of the acceleration of the particles.[5]
(b)It is given instead that the plane BC is rough. A force of magnitude 3N is applied to Q directly up the plane along a line of greatest slope of the plane. Find the least value of the coefficient of friction between Q and the plane BC for which the particles remain at rest.[5]

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Mark scheme

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

[M1] Attempt to use Newton's Second law
[A1] For any one equation [e.g. For P: 0.3g sin 60 - T = 0.3a]
[A1] For any second equation [e.g. For Q: T + 0.2g sin 30 = 0.2a]
[M1] For solving for a or T
[A1] Magnitude of acceleration = 7.20 ms⁻² Tension = 0.439 N
[B1] R = 0.2g cos 30
[B1] Equilibrium for P [0.3g sin 60 - T = 0 or T = 3√3 / 2 or T = 2.598...]
[M1] Equilibrium for Q on the point of moving down [T + 0.2g sin 30 - F - 3 = 0]
[M1] Use of F = µR
[A1] µ = 0.345

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