| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Prove SHM and find period: horizontal or non-standard geometry |
| Difficulty | Challenging +1.2 This is a standard M3/Further Mechanics SHM question requiring resolution of forces, proving SHM form (a = -ω²x), and applying standard SHM formulas. While it involves multiple parts and requires careful geometric reasoning to resolve the force component, the techniques are well-practiced and follow a predictable structure. The question is moderately harder than average due to the force resolution step and multi-part nature, but doesn't require novel insight beyond standard M3 methods. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(R(\rightarrow)\): \(0.2\ddot{x} = -5y\cos\theta\) | M1 A1 | |
| \(\cos\theta = \frac{x}{y}\) | M1 | |
| \(\Rightarrow \ddot{x} = -25x\) | A1 | |
| \(\Rightarrow\) SHM period \(= \frac{2\pi}{5}\) | A1 | (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(d = 2\); max speed \(= d\omega = 2 \times 5 = 10\) m s\(^{-1}\) | M1 A1 | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x = 2\cos 5t\) | M1 | |
| Distance 3 m from start \(\Rightarrow x = -1\) | B1 | |
| \(\cos 5t = -\frac{1}{2}\) | ||
| \(\Rightarrow 5t = \frac{2\pi}{3}\), \(t = \frac{2\pi}{15}\) s | M1 A1 | (4) |
## Question 4:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $R(\rightarrow)$: $0.2\ddot{x} = -5y\cos\theta$ | M1 A1 | |
| $\cos\theta = \frac{x}{y}$ | M1 | |
| $\Rightarrow \ddot{x} = -25x$ | A1 | |
| $\Rightarrow$ SHM period $= \frac{2\pi}{5}$ | A1 | (5) |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $d = 2$; max speed $= d\omega = 2 \times 5 = 10$ m s$^{-1}$ | M1 A1 | (2) |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = 2\cos 5t$ | M1 | |
| Distance 3 m from start $\Rightarrow x = -1$ | B1 | |
| $\cos 5t = -\frac{1}{2}$ | | |
| $\Rightarrow 5t = \frac{2\pi}{3}$, $t = \frac{2\pi}{15}$ s | M1 A1 | (4) |
**(11 marks)**
---4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{45c51316-7d58-4c16-9b5f-1d7421060a88-4_332_1056_251_459}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}
A small smooth bead $B$ of mass 0.2 kg is threaded on a smooth horizontal wire. The point $A$ is on the same horizontal level as the wire and at a perpendicular distance $d$ from the wire. The point $O$ is the point on the wire nearest to $A$, as shown in Fig. 2. The bead experiences a force of magnitude $5 ( A B )$ newtons in the direction $B A$ towards $A$. Initially $B$ is at rest with $O B = 2 \mathrm {~m}$.
\begin{enumerate}[label=(\alph*)]
\item Prove that $B$ moves with simple harmonic motion about $O$, with period $\frac { 2 \pi } { 5 } \mathrm {~s}$.
\item Find the greatest speed of $B$ in the motion.
\item Find the time when $B$ has first moved a distance 3 m from its initial position.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q4 [11]}}