A-LevelMathematicsMechanics9709/42 · Oct/Nov 2020 · Q89 Marks

Two particles A and B, of masses 0.3 kg and 0.5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to a horizontal plane and to the top of an inclined plane. The particles are initially at rest with A on the horizontal plane and B on the inclined plane, which makes an angle of 30° with the horizontal. The string is taut and B can move on a line of greatest slope of the inclined plane. A force of magnitude 3.5 N is applied to B acting down the plane (see diagram).

(a)Given that both planes are smooth, find the tension in the string and the acceleration of B.[5]
(b)It is given instead that the two planes are rough. When each particle has moved a distance of 0.6 m from rest, the total amount of work done against friction is 1.1 J. Use an energy method to find the speed of B when it has moved this distance down the plane. [You should assume that the string is sufficiently long so that A does not hit the pulley when it moves 0.6 m.][4]

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 2020). The mark or mark type is shown in brackets.

[5] (a) For A: T=0.3a
[5] (a) For applying Newton's 2nd law for either particle A or to particle B or to the system. Correct number of terms.
[5] (a) For B: 3.5 + 0.5g sin 30 – T = 0.5a
[5] (a) Two correct equations
[5] (a) System: 3.5 + 0.5g sin 30 = (0.3 + 0.5)a
[5] (a) For solving either for T or for a
[5] (a) a = 7.5 ms⁻²
[5] (a) T = 2.25 N
[4] (b) 0.5g sin 30 × 0.6 [= 1.5]
[4] (b) PE loss by B
[4] (b) Apply the work-energy equation to the system
[4] (b) 5 relevant terms, their PE for 0.5 kg, WD by 3.5 N, WD against friction and two relevant KE terms.
[4] (b) 0.5g sin 30 × 0.6 + 3.5 × 0.6 = ½ × 0.8 × v² + 1.1
[4] (b) v = 2.5 ms⁻¹

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