| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2008 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Displacement and velocity at given time |
| Difficulty | Standard +0.3 This is a standard SHM question requiring application of standard formulas (x = a sin(ωt + ε), v = ω√(a² - x²)) with straightforward setup. The amplitude and period are directly given through the geometry, and all three parts follow routine procedures without requiring novel insight or complex manipulation. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x4.10g Damped oscillations: model and interpret |
| Answer | Marks |
|---|---|
| \(T = 3 = \frac{2\pi}{\omega} \therefore \omega = \frac{2\pi}{3}\) | M1A1 |
| \(u^2 = \omega^2(a^2 - x^2)\); \(a = 0.12\), \(u^2 = a^2\omega^2, u = 0.12 \times \omega = 0.251 \text{ ms}^{-1}\) (0.25 m s\(^{-1}\)) | M1 A1 (4) |
| Answer | Marks |
|---|---|
| Time from \(O \to A \to O = 1.5s \therefore t = 0.5\) | B1 |
| \(x = a\sin\omega t \Rightarrow OP = 0.12\sin\left(\frac{\pi}{3}\right)\) | M1A1 |
| Distance from \(B\) is \(0.12 - OP = 0.12 - 0.104... = 0.016m\) | M1A1 (5) |
| Answer | Marks |
|---|---|
| \(v^2 = \omega^2(a^2 - x^2)\) | M1 |
| \(v = \frac{2\pi}{3}\sqrt{0.12^2 - 0.104...^2} = \frac{2\pi}{3} \times 0.0598 = 0.13 \text{ ms}^{-1}\) | A1 (2) |
**Part (a):**
$T = 3 = \frac{2\pi}{\omega} \therefore \omega = \frac{2\pi}{3}$ | M1A1 |
$u^2 = \omega^2(a^2 - x^2)$; $a = 0.12$, $u^2 = a^2\omega^2, u = 0.12 \times \omega = 0.251 \text{ ms}^{-1}$ (0.25 m s$^{-1}$) | M1 A1 (4) |
**Part (b):**
Time from $O \to A \to O = 1.5s \therefore t = 0.5$ | B1 |
$x = a\sin\omega t \Rightarrow OP = 0.12\sin\left(\frac{\pi}{3}\right)$ | M1A1 |
Distance from $B$ is $0.12 - OP = 0.12 - 0.104... = 0.016m$ | M1A1 (5) |
**Part (c):**
$v^2 = \omega^2(a^2 - x^2)$ | M1 |
$v = \frac{2\pi}{3}\sqrt{0.12^2 - 0.104...^2} = \frac{2\pi}{3} \times 0.0598 = 0.13 \text{ ms}^{-1}$ | A1 (2) |
**Total: 11 Marks**
---2. A particle $P$ moves with simple harmonic motion and comes to rest at two points $A$ and $B$ which are 0.24 m apart on a horizontal line. The time for $P$ to travel from $A$ to $B$ is 1.5 s . The midpoint of $A B$ is $O$. At time $t = 0 , P$ is moving through $O$, towards $A$, with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $u$.
\item Find the distance of $P$ from $B$ when $t = 2 \mathrm {~s}$.
\item Find the speed of $P$ when $t = 2 \mathrm {~s}$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2008 Q2 [11]}}