A-LevelMathematicsMechanics9709/52 · Oct/Nov 2018 · Q27 Marks

A uniform solid object is made by attaching a cone to a cylinder so that the circumferences of the base of the cone and a plane face of the cylinder coincide. The cone and the cylinder each have radius 0.3 m and height 0.4 m.

(i)Calculate the distance of the centre of mass of the object from the vertex of the cone. [The volume of a cone is πr²h.][4]
(ii)The object has weight WN and is placed with its plane circular face on a rough horizontal surface. A force of magnitude kW N acting at 30° to the upward vertical is applied to the vertex of the cone. The object does not slip. Find the greatest possible value of k for which the object does not topple.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2018). The mark or mark type is shown in brackets.

[M1] Attempt to take moments about the vertex of the cone
[A1] (π × 0.3² × 0.4/3) × (3 × 0.4/4) + (π × 0.3² × 0.4 × (0.4 + 0.2))
[A1] (π × 0.3² × 0.4/3 + π × 0.3² × 0.4)x̄
[A1] x̄ = 0.525 m
[M1] Attempt to take moments about a point on the circumference of the base of the cone
[A1] kWcos30 × 0.3 + kWsin30 × 0.8 = 0.3W
[A1] k = 0.455

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