M2 January 2007 Q3
3.
Figure 1 shows a template \(T\) made by removing a circular disc, of centre \(X\) and radius 8 cm, from a uniform circular lamina, of centre \(O\) and radius 24 cm. The point \(X\) lies on the diameter \(AOB\) of the lamina and \(AX = 16\) cm. The centre of mass of \(T\) is at the point \(G\).
(a) Find \(AG\). (6)
The template \(T\) is free to rotate about a smooth fixed horizontal axis, perpendicular to the plane of \(T\), which passes through the mid-point of \(OB\). A small stud of mass \(\tfrac{1}{4}m\) is fixed at \(B\), and \(T\) and the stud are in equilibrium with \(AB\) horizontal. Modelling the stud as a particle,
(b) find the mass of \(T\) in terms of \(m\). (4)
| Scheme | Marks | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| B1, B1ft | ||||||||||||
| M(\(A\)) \(9 \times 24 = 16 \times 1 + 8\bar{x}\) | M1* A1 | ||||||||||||
| \(\bar{x} = 25\) (cm) exact | DM1* A1 | ||||||||||||
| (6) |
| Scheme | Marks |
|---|---|
| M(axis) \(11M = 12 \times \dfrac{1}{4}m\) ft their \(\bar{x}\) | M1 † A1ft |
| \(\left((36 - \bar{x})M = 12 \times \dfrac{1}{4}m\right)\) | |
| \(M = \dfrac{3}{11}m\) (o.e.e.) | DM1 † A1 |
| (4) | |
| (10 marks) |