M3 January 2007 Q2

Figure 1: cone hanging from a string at A on the rim of its base, base radius a, semi-vertical angle alpha, axis at theta degrees to the upward vertical — Figure 1

A uniform solid right circular cone has base radius \(a\) and semi-vertical angle \(\alpha\), where \(\tan\alpha = \tfrac{1}{3}\). The cone is freely suspended by a string attached at a point \(A\) on the rim of its base, and hangs in equilibrium with its axis of symmetry making an angle of \(\theta^\circ\) with the upward vertical, as shown in Figure 1.

Find, to one decimal place, the value of \(\theta\).

Diagram: cone with A vertically above the centre of mass G, base radius a, distance h from the base to G — Mark scheme

Scheme	Marks
Height of cone \(= \dfrac{a}{\tan\alpha} = 3a\)	M1 A1
Hence \(h = \tfrac{3}{4}a\)	↓ M1
\(\tan\theta = \dfrac{a}{\frac{3}{4}a} = \dfrac{4}{3} \Rightarrow \theta = 53.1^\circ\)	↓ M1 A1
	(5)
	(5 marks)

Notes