| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2012 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Speed at given displacement |
| Difficulty | Standard +0.3 This is a straightforward SHM question requiring standard formula application: finding ω from period, then using a = -ω²x for acceleration and v² = ω²(a² - x²) for speed, plus basic time calculation. All steps are routine M3 techniques with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(T = \frac{2\pi}{\omega} = \frac{2\pi}{3}\), \(\omega = 3\) | B1 | |
| \( | a | = \omega^2 x = 9 \times 0.2 = 1.8 \text{ ms}^{-2}\) towards \(C\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(v^2 = \omega^2(a^2 - x^2) = 9(0.25 - 0.04) = 1.89\) | M1 | |
| \(v = 1.37 \text{ ms}^{-1}\) | A1 | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x = 0.5\sin 3t = 0.2\) | M1 A1ft | |
| \(t = \frac{1}{3}\sin^{-1} 0.4 \approx 0.137 \text{ s}\) | A1 | (3) Total: 8 |
## Question 2:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $T = \frac{2\pi}{\omega} = \frac{2\pi}{3}$, $\omega = 3$ | B1 | |
| $|a| = \omega^2 x = 9 \times 0.2 = 1.8 \text{ ms}^{-2}$ towards $C$ | M1 A1 | (3) |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $v^2 = \omega^2(a^2 - x^2) = 9(0.25 - 0.04) = 1.89$ | M1 | |
| $v = 1.37 \text{ ms}^{-1}$ | A1 | (2) |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = 0.5\sin 3t = 0.2$ | M1 A1ft | |
| $t = \frac{1}{3}\sin^{-1} 0.4 \approx 0.137 \text{ s}$ | A1 | (3) Total: 8 |
---2. A particle $P$ is moving in a straight line with simple harmonic motion. The centre of the oscillation is the fixed point $C$, the amplitude of the oscillation is 0.5 m and the time to complete one oscillation is $\frac { 2 \pi } { 3 }$ seconds. The point $A$ is on the path of $P$ and 0.2 m from $C$. Find
\begin{enumerate}[label=(\alph*)]
\item the magnitude and direction of the acceleration of $P$ when it passes through $A$,
\item the speed of $P$ when it passes through $A$,
\item the time $P$ takes to move directly from $C$ to $A$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2012 Q2 [8]}}