M3 June 2007 Q2
2. An open container \(C\) is modelled as a thin uniform hollow cylinder of radius \(h\) and height \(h\) with a base but no lid. The centre of the base is \(O\).
(a) Show that the distance of the centre of mass of \(C\) from \(O\) is \(\dfrac{1}{3}h\). (5)
The container is filled with uniform liquid. Given that the mass of the container is \(M\) and the mass of the liquid is \(M\),
(b) find the distance of the centre of mass of the filled container from \(O\). (5)
| Scheme | Marks | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| B1 B1 | ||||||||||||
| \(3\pi h^2 \times \bar{y} = 2\pi h^2 \times \dfrac{h}{2}\) | M1 A1 | ||||||||||||
| Leading to \(\bar{y} = \dfrac{1}{3}h\) * cso | A1 | ||||||||||||
| (5) |
| Scheme | Marks | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| B1 B1 | ||||||||||||
| \(2M \times \bar{y} = M \times \dfrac{h}{2} + M \times \dfrac{h}{3}\) | M1 A1 | ||||||||||||
| \(\bar{y} = \dfrac{5}{12}h\) | A1 | ||||||||||||
| (5) | |||||||||||||
| (10 marks) |