5 The triangular region shown below is rotated through \(360 ^ { \circ }\) around the \(x\)-axis, to form a solid cone.
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The coordinates of the vertices of the triangle are \(( 0,0 ) , ( 8,0 )\) and \(( 0,4 )\).
All units are in centimetres.
5
- State an assumption that you should make about the cone in order to find the position of its centre of mass.
5
- Using integration, prove that the centre of mass of the cone is 2 cm from its plane face.
5 - The cone is placed with its plane face on a rough board. One end of the board is lifted so that the angle between the board and the horizontal is gradually increased. Eventually the cone topples without sliding.
5
- Find the angle between the board and the horizontal when the cone topples, giving your answer to the nearest degree.
5
- (ii) Find the range of possible values for the coefficient of friction between the cone and the board.