A-LevelMathematicsMechanics9709/43 · May/June 2020 · Q712 Marks

Two particles A and B, of masses 3m kg and 2m kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a plane. The plane is inclined at an angle θ to the horizontal. A lies on the plane and B hangs vertically, 0.8 m above the floor, which is horizontal. The string between A and the pulley is parallel to a line of greatest slope of the plane (see diagram). Initially A and B are at rest.

(a)Given that the plane is smooth, find the value of θ for which A remains at rest.[3]
(b)Show that the coefficient of friction between A and the plane is √3/10.[5]
(c)When B reaches the floor it comes to rest. Find the length of time after B reaches the floor for which A is moving up the plane. [You may assume that A does not reach the pulley.][4]

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Mark scheme

Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.

[B1] T – 2mg = 0
[M1] 3mg sin θ – T = 0 (for resolving forces parallel to the plane and solving for θ)
[A1] θ = 41.8 (41.810...)
[B1] R = 3mgcos30
[M1] Use of F = μR
[M1] 2mg – T = 0.1 × 2m OR T – 3mg sin30 – μ × 3mg cos30 = 0.1 × 3m
[M1] 2mg – 0.2m – 3mg sin30 – μ × 3mg cos30 = 0.1 × 3m
[A1] μ = √3/10
[M1] v² = 0 + 2 × 0.1 × 0.8 (v = 0.4)
[M1] -3mg sin30 – μ × 3mg cos30 = 3ma (a = -6.5)
[M1] 0 = 0.4 – 6.5t
[A1] t = 0.4 / 6.5 = 0.0615 s

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