M3 June 2009 Q3
3.
A particle \(P\) of mass \(m\) moves on the smooth inner surface of a hemispherical bowl of radius \(r\). The bowl is fixed with its rim horizontal as shown in Figure 2. The particle moves with constant angular speed \(\sqrt{\left(\dfrac{3g}{2r}\right)}\) in a horizontal circle at depth \(d\) below the centre of the bowl.
(a) Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction of the bowl on \(P\). (4)
(b) Find \(d\) in terms of \(r\). (4)
| Scheme | Marks |
|---|---|
| \(\leftrightarrow\) \(R\sin\theta = mx\omega^2\) | M1 A1 |
| \(R \times \dfrac{x}{r} = mx \times \dfrac{3g}{2r}\) | M1 |
| \(R = \dfrac{3mg}{2}\) | A1 |
| Scheme | Marks |
|---|---|
| \(\updownarrow\) \(R\cos\theta = mg\) | M1 A1 |
| \(\dfrac{3mg}{2} \times \dfrac{d}{r} = mg\) | M1 |
| \(d = \dfrac{2}{3}r\) | A1 |
| (8 marks) |