M3 June 2005 Q4
4. A particle \(P\) of mass \(m\) moves on the smooth inner surface of a spherical bowl of internal radius \(r\). The particle moves with constant angular speed in a horizontal circle, which is at a depth \(\tfrac{1}{2}r\) below the centre of the bowl.
(a) Find the normal reaction of the bowl on \(P\). (4)
(b) Find the time for \(P\) to complete one revolution of its circular path. (6)
| Scheme | Marks |
|---|---|
| \(\sin\theta = \dfrac{\frac{1}{2}r}{r} = \dfrac{1}{2}\ \ (\Rightarrow \theta = 30^\circ)\) | B1 |
| \(\uparrow\) \(R\sin\theta = mg\) | M1 A1 |
| \(R = 2mg\) | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\rightarrow\) \(R\cos\theta = mx\omega^2\) | M1 A1 |
| \(= m(r\cos\theta)\omega^2\) | A1 |
| \(\omega = \left(\dfrac{2g}{r}\right)^{\frac{1}{2}}\) | A1 |
| \(T = \dfrac{2\pi}{\omega} = 2\pi\left(\dfrac{r}{2g}\right)^{\frac{1}{2}}\) or exact equivalent | M1 A1 |
| (6) | |
| (10 marks) |
Notes
Note: \(x = \dfrac{\sqrt{3}}{2}r\)