A-LevelMathematicsMechanics9709/43 · Oct/Nov 2020 · Q711 Marks

As shown in the diagram, particles A and B of masses 2 kg and 3 kg respectively are attached to the ends of a light inextensible string. The string passes over a small fixed smooth pulley which is attached to the top of two inclined planes. Particle A is on plane P, which is inclined at an angle of 10° to the horizontal. Particle B is on plane Q, which is inclined at an angle of 20° to the horizontal. The string is taut, and the two parts of the string are parallel to lines of greatest slope of their respective planes. [Figure 7.1]

(a)It is given that plane P is smooth, plane Q is rough, and the particles are in limiting equilibrium. Find the coefficient of friction between particle B and plane Q.[5]
(b)It is given instead that both planes are smooth and that the particles are released from rest at the same horizontal level. Find the time taken until the difference in the vertical height of the particles is 1 m. [You should assume that this occurs before A reaches the pulley or B reaches the bottom of plane Q.][6]

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Mark scheme

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

[1] Resolve forces parallel to plane P for particle A or parallel to plane Q for Particle B
[1] T = 2g sin 10 and 3g sin 20 = F + T
[1] Resolving forces perpendicular to plane Q for particle B
[1] Using μ = F/R
[1] μ = 0.241 (=0.2407...)
[1] For applying Newton's second law to either A or to B or to the system
[1] For applying Newton's second law to the second particle and/or solving for a
[1] a = 1.3575...
[1] Using expressions for height change of each particle after each moves a distance x along the plane, to obtain equation in x
[1] For using s = ut + ½at² for either particle with s = x, u = 0 and using their a (= 1.3575)
[1] t = 1.69

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