M2 January 2008 Q4
4.
A set square \(S\) is made by removing a circle of centre \(O\) and radius 3 cm from a triangular piece of wood. The piece of wood is modelled as a uniform triangular lamina \(ABC\), with \(\angle ABC = 90^\circ\), \(AB = 12\) cm and \(BC = 21\) cm. The point \(O\) is 5 cm from \(AB\) and 5 cm from \(BC\), as shown in Figure 1.
(a) Find the distance of the centre of mass of \(S\) from
(i) \(AB\),
(ii) \(BC\). (9)
The set square is freely suspended from \(C\) and hangs in equilibrium.
(b) Find, to the nearest degree, the angle between \(CB\) and the vertical. (3)
| Scheme | Marks |
|---|---|
| \(\begin{array}{lccc} & \text{Triangle} & \text{Circle} & S\\ \text{Mass ratio} & 126 & 9\pi & 126 - 9\pi\\ & & (28.3) & (97.7)\\ \bar{x} & 7 & 5 & \bar{x}\\ \bar{y} & 4 & 5 & \bar{y}\end{array}\) | B1 B1ft |
| 4, 7 seen | B1 |
| \(126 \times 7 = 9\pi \times 5 + (126 - 9\pi) \times \bar{x}\) ft their table values | M1 A1ft |
| \(\bar{x} \approx 7.58\ \ \left(\dfrac{882 - 45\pi}{126 - 9\pi}\right)\) awrt 7.6 | A1 |
| \(126 \times 4 = 9\pi \times 5 + (126 - 9\pi) \times \bar{y}\) ft their table values | M1 A1ft |
| \(\bar{y} \approx 3.71\ \ \left(\dfrac{504 - 45\pi}{126 - 9\pi}\right)\) awrt 3.7 | A1 |
| (9) |
| Scheme | Marks |
|---|---|
| \(\tan\theta = \dfrac{\bar{y}}{21 - \bar{x}}\) ft their \(\bar{x}\), \(\bar{y}\) | M1 A1ft |
| \(\theta \approx 15^\circ\) | A1 |
| (3) | |
| (12 marks) |