M2 January 2005 Q2

Figure 2: rectangle ABCD with AB = 20 cm and BC = 10 cm, circle of radius 3 cm and centre O removed, O 5 cm from AB and 6 cm from AD — Figure 2

Figure 2 shows a metal plate that is made by removing a circle of centre \(O\) and radius 3 cm from a uniform rectangular lamina \(ABCD\), where \(AB = 20\) cm and \(BC = 10\) cm. The point \(O\) is 5 cm from both \(AB\) and \(CD\) and is 6 cm from \(AD\).

(a) Calculate, to 3 significant figures, the distance of the centre of mass of the plate from \(AD\). (5)

The plate is freely suspended from \(A\) and hangs in equilibrium.

(b) Calculate, to the nearest degree, the angle between \(AB\) and the vertical. (3)

Mark scheme (a)

Scheme

Marks

	circle	rectangle	plate
Mass ratios	\(9\pi\)	200;	\(200 - 9\pi\)
Centres of mass	6	10	\(\bar{x}\)

B1; B1ft
B1

\(9\pi \times 6 + (200 - 9\pi)\bar{x} = 200 \times 10\)

\(\bar{x} \approx 10.7\) (cm) cao

(5)

Scheme	Marks
\(\tan\theta = \dfrac{5}{10.7}\) ft their \(\bar{x}\)	M1 A1ft
\(\theta \approx 25^\circ\) cao	A1
	(3)
	(8 marks)