| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2009 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Tidal/harbour water level SHM |
| Difficulty | Standard +0.3 This is a standard SHM application question requiring identification of amplitude and period from context, then applying standard velocity formula and solving for time. The setup is slightly more involved than basic SHM questions due to the real-world context and needing to determine the equilibrium position, but the mathematical techniques are routine for M3 students who have practiced SHM problems. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(a = 8\) | B1 | |
| \(T = \frac{25}{2} = \frac{2\pi}{\omega} \Rightarrow \omega = \frac{4\pi}{25} (\approx 0.502...)\) | M1 A1 | |
| \(v^2 = \omega^2(a^2 - x^2) \Rightarrow v^2 = \left(\frac{4\pi}{25}\right)^2(8^2 - 3^2)\) | M1 A1ft | ft their \(a\), \(\omega\) |
| \(v = \frac{4\pi}{25}\sqrt{55} \approx 3.7 \text{ (m h}^{-1})\) | M1 A1 | awrt 3.7 (7) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(x = a\cos\omega t \Rightarrow 3 = 8\cos\left(\frac{4\pi}{25}t\right)\) | M1 A1ft | ft their \(a\), \(\omega\) |
| \(t \approx 2.3602...\) | M1 | |
| time is 12:22 | A1 | (4) [11] |
## Question 4:
### Part (a):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $a = 8$ | B1 | |
| $T = \frac{25}{2} = \frac{2\pi}{\omega} \Rightarrow \omega = \frac{4\pi}{25} (\approx 0.502...)$ | M1 A1 | |
| $v^2 = \omega^2(a^2 - x^2) \Rightarrow v^2 = \left(\frac{4\pi}{25}\right)^2(8^2 - 3^2)$ | M1 A1ft | ft their $a$, $\omega$ |
| $v = \frac{4\pi}{25}\sqrt{55} \approx 3.7 \text{ (m h}^{-1})$ | M1 A1 | awrt 3.7 **(7)** |
### Part (b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $x = a\cos\omega t \Rightarrow 3 = 8\cos\left(\frac{4\pi}{25}t\right)$ | M1 A1ft | ft their $a$, $\omega$ |
| $t \approx 2.3602...$ | M1 | |
| time is 12:22 | A1 | **(4) [11]** |
---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 1000 hours and the next time that this minimum depth occurs is at 2230 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
\begin{enumerate}[label=(\alph*)]
\item the speed, in metres per hour, at which the water level is rising when it reaches the shellfish,
\item the earliest time after 1000 hours on this day at which the water reaches the shellfish.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2009 Q4 [11]}}