OCR A Level: Mechanics 2 - Sliding And Toppling [June 2011 Q7]

A uniform solid cone of height $0.8m$ and semi-vertical angle $60°$ lies with its curved surface on a horizontal plane. The point $P$ on the circumference of the base is in contact with the plane. $V$ is the vertex of the cone and $PQ$ is a diameter of its base. The weight of the cone is $550 \text{N}$ . A force of magnitude $F \text{N}$ and line of action $PQ$ is applied to the base of the cone (see Fig. 1). The cone topples about $V$ without sliding.

$(\text{i})$ Calculate the least possible value of $F$ .

The force of magnitude $F \text{N}$ is removed and an increasing force of magnitude $T \text{N}$ acting upwards in the vertical plane of symmetry of the cone and perpendicular to $PQ$ is applied to the cone at $Q$ (see Fig. 2). The coefficient of friction between the cone and the horizontal plane is $\mu$ .

$(\text{ii})$ Given that the cone slides before it topples about $P$ , calculate the greatest possible value for $\mu$ .

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Input the nearest integer to $1000 \mu$ as your answer.

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There are 4 marks available for part (i) and 10 marks for part (ii).

In total, this question is worth 19.4% of all available marks in the paper.

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This is part of the set OCR A Level Problems .