M3 January 2009 Q4
4. A small shellfish is attached to a wall in a harbour. The rise and fall of the water level is modelled as simple harmonic motion and the shellfish as a particle. On a particular day the minimum depth of water occurs at 10 00 hours and the next time that this minimum depth occurs is at 22 30 hours. The shellfish is fixed in a position 5 m above the level of the minimum depth of the water and 11 m below the level of the maximum depth of the water. Find
(a) the speed, in metres per hour, at which the water level is rising when it reaches the shellfish, (7)
(b) the earliest time after 10 00 hours on this day at which the water reaches the shellfish. (4)
| Scheme | Marks |
|---|---|
| \(a = 8\) | B1 |
| \(T = \dfrac{25}{2} = \dfrac{2\pi}{\omega} \Rightarrow \omega = \dfrac{4\pi}{25}\ (\approx 0.502\ \ldots)\) | M1 A1 |
| \(v^2 = \omega^2\left(a^2 - x^2\right) \Rightarrow v^2 = \left(\dfrac{4\pi}{25}\right)^2\left(8^2 - 3^2\right)\) ft their \(a,\ \omega\) | M1 A1ft |
| \(v = \dfrac{4\pi}{25}\sqrt{55} \approx 3.7\ \left(\text{m h}^{-1}\right)\) awrt 3.7 | M1 A1 |
| (7) |
| Scheme | Marks |
|---|---|
| \(x = a\cos\omega t \Rightarrow 3 = 8\cos\left(\dfrac{4\pi}{25}t\right)\) ft their \(a,\ \omega\) | M1 A1ft |
| \(t \approx 2.3602\ \ldots\) | M1 |
| time is 12 22 | A1 |
| (4) | |
| (11 marks) |