M3 June 2006 Q4
4.
A hollow cone, of base radius \(3a\) and height \(4a\), is fixed with its axis vertical and vertex \(V\) downwards, as shown in Figure 1. A particle moves in a horizontal circle with centre \(C\), on the smooth inner surface of the cone with constant angular speed \(\sqrt{\dfrac{8g}{9a}}\).
Find the height of \(C\) above \(V\). (11)
| Scheme | Marks |
|---|---|
| \(\tan\alpha = \dfrac{3}{4}\) or equivalent | B1 |
| \(\tan\alpha = \dfrac{r}{h}\) or \(\dfrac{r}{h} = \dfrac{3a}{4a}\) | B1 |
| \(R(\uparrow)\) \(R\sin\alpha = mg\) \(\left(R = \dfrac{5}{3}mg\right)\) | M1 A1 |
| \(R(\leftarrow)\) \(R\cos\alpha = mr\omega^2\) | M1 A1 |
| \(= mr \times \dfrac{8g}{9a}\) \(\left(R = \dfrac{10mrg}{9a}\right)\) | A1 |
| Eliminating \(R\) \(\tan\alpha = \dfrac{9a}{8r}\) \(\left(\dfrac{5}{3}mg = \dfrac{10mrg}{9a}\right)\) | M1 A1 |
| \(\left(\dfrac{3}{4} = \dfrac{9a}{8r} \Rightarrow r = \dfrac{3}{2}a\right)\) | |
| \(h = \dfrac{r}{\tan\alpha} = \dfrac{3a}{2} \times \dfrac{4}{3} = 2a\) | M1 A1 |
| (11) | |
| (11 marks) |
Notes
The scheme brackets the marks from \(R\cos\alpha = mr\omega^2\) to eliminating \(R\).