M3 January 2008 Q3

Figure 1: frustum of a cone with base radius 2r and top radius r; the full cone has height 2h and the removed cone above has height h — Figure 1

A uniform solid \(S\) is formed by taking a uniform solid right circular cone, of base radius \(2r\) and height \(2h\), and removing the cone, with base radius \(r\) and height \(h\), which has the same vertex as the original cone, as shown in Figure 1.

(a) Show that the distance of the centre of mass of \(S\) from its larger plane face is \(\tfrac{11}{28}h\). (5)

The solid \(S\) lies with its larger plane face on a rough table which is inclined at an angle \(\theta^\circ\) to the horizontal. The table is sufficiently rough to prevent \(S\) from slipping. Given that \(h = 2r\),

(b) find the greatest value of \(\theta\) for which \(S\) does not topple. (3)

Mark scheme (a)

Scheme

Marks

	Large cone	small cone	\(S\)
Vol.	\(\tfrac{1}{3}\pi(2r)^2(2h)\)	\(\tfrac{1}{3}\pi r^2h\)	\(\tfrac{7}{3}\pi r^2h\)
C of M	\(\tfrac{1}{2}h\),	\(\tfrac{5}{4}h\)	\(\bar{x}\)

Vol.: (accept ratios 8 : 1 : 7) C of M: (or equivalent)

B1
B1, B1

\(\tfrac{8}{3}\pi r^2h.\tfrac{1}{2}h - \tfrac{1}{3}\pi r^2h.\tfrac{5}{4}h = \tfrac{7}{3}\pi r^2h.\bar{x}\) or equivalent

\(\rightarrow \bar{x} = \tfrac{11}{28}h\) *

(5)

Notes

Centres of mass may be measured from another point (e.g. centre of small circle, or vertex). The Method mark will then require a complete method (Moments and subtraction) to give required value for \(\bar{x}\)). However B marks can be awarded for correct values if the candidate makes the working clear.

Mark scheme (b)

Scheme	Marks
\(\tan\theta = \dfrac{2r}{\bar{x}} = \dfrac{2r}{\frac{11}{28}h},\ = \dfrac{2r}{\frac{11}{14}r} = \dfrac{28}{11}\)	M1, A1
\(\theta \approx 68.6^\circ\) or 1.20 radians	A1
	(3)
	(8 marks)

Notes

(Special case – obtains complement by using \(\tan\theta = \dfrac{\bar{x}}{2r}\) giving 21.4\(^\circ\) or .374 radians M1A0A0)

(Corrected from the printed mark scheme: the special case is printed as \(\tan\theta = \dfrac{2r}{\bar{x}}\).)