| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Time to travel between positions |
| Difficulty | Standard +0.3 This is a standard SHM question requiring knowledge of amplitude-period relationships and position-time equations. Part (a) is routine (v_max = ωa). Part (b) involves solving x = a cos(ωt) for two positions and finding the time difference, which is a common textbook exercise requiring careful setup but no novel insight. The 6 marks reflect moderate working rather than conceptual difficulty. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x |
| Answer | Marks | Guidance |
|---|---|---|
| (a) amplitude \(= \frac{1}{2} \times 8 = 4\) m | B1 | |
| period \(= \frac{2\pi}{\omega} = 12 \therefore \omega = \frac{\pi}{6}\) | B1 | |
| \(v_{\max} = a\omega = 4 \times \frac{\pi}{6} = \frac{2\pi}{3}\) ms\(^{-1}\) | M1 A1 | |
| (b) \(x = a \sin \omega t\) | M1 | |
| at P, \(1 = 4 \sin \omega t \therefore \frac{\pi}{6}t = 0.2527, t = 0.4826\) | M1 A1 | |
| at Q, \(2 = 4 \sin \omega t \therefore \frac{\pi}{6}t = \frac{\pi}{6}, t = 1\) | M1 A1 | |
| \(\therefore\) time between \(= 1.48\) s (3sf) | A1 | (10) |
(a) amplitude $= \frac{1}{2} \times 8 = 4$ m | B1
period $= \frac{2\pi}{\omega} = 12 \therefore \omega = \frac{\pi}{6}$ | B1
$v_{\max} = a\omega = 4 \times \frac{\pi}{6} = \frac{2\pi}{3}$ ms$^{-1}$ | M1 A1
(b) $x = a \sin \omega t$ | M1
at P, $1 = 4 \sin \omega t \therefore \frac{\pi}{6}t = 0.2527, t = 0.4826$ | M1 A1
at Q, $2 = 4 \sin \omega t \therefore \frac{\pi}{6}t = \frac{\pi}{6}, t = 1$ | M1 A1
$\therefore$ time between $= 1.48$ s (3sf) | A1 | (10)
---3. A particle is performing simple harmonic motion along a straight line between the points $A$ and $B$ where $A B = 8 \mathrm {~m}$. The period of the motion is 12 seconds.
\begin{enumerate}[label=(\alph*)]
\item Find the maximum speed of the particle in terms of $\pi$.
The points $P$ and $Q$ are on the line $A B$ at distances of 3 m and 6 m respectively from $A$.
\item Find, correct to 3 significant figures, the time it takes for the particle to travel directly from $P$ to $Q$.\\
(6 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q3 [10]}}