| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | String becomes slack timing |
| Difficulty | Challenging +1.8 This M3 question combines elastic strings, friction, and SHM in a multi-stage problem requiring careful force analysis and energy methods. Part (a) is routine equilibrium. Part (b) requires showing SHM about a shifted equilibrium point (non-trivial). Parts (c) and (d) demand sophisticated understanding of when the string goes slack, applying SHM equations with specific initial conditions, then switching to energy methods—this extended reasoning with multiple conceptual transitions places it well above average difficulty. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x6.02g Hooke's law: T = k*x or T = lambda*x/l6.02h Elastic PE: 1/2 k x^2 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Eqn.: \(T = F\) | \(\frac{\lambda}{4} \cdot \frac{\lambda}{4} - \mu mg\) | \(\mu = \frac{1}{4}\) |
| (b) At dist. \(x\), \(T - \mu mg = -mx\) | \(\frac{mg}{4}\left(\frac{1}{4}+x\right) - \frac{1}{4}mg = -mx\) | M1 A1 A1 |
| \(x = -\frac{x}{x}\) | S.H.M. with \(\omega^2 = \frac{k}{l}\) | A1 |
| (c) Amplitude = \(\frac{3l}{4}\) | \(x = \frac{3l}{4}\cos\omega t\) | \(x = -\frac{l}{4}\); \(t = \frac{1}{\omega}\arccos\left(-\frac{1}{3}\right)\) |
| \(= \frac{l}{g}\left(\frac{\pi}{2} + \arcsin\left(\frac{1}{1}\right)\right)s\) | M1 A1 | |
| (d) At nat. length, \(v^2 = \frac{l}{t}\left(\frac{9l}{16} - \frac{l}{16}\right) = \frac{gl}{2}\) | At \(O\), \(v^2 = \frac{gl}{2} - 2\frac{l}{k}\), \(v=0\) | M1 A1 M1 A1 |
(a) Eqn.: $T = F$ | $\frac{\lambda}{4} \cdot \frac{\lambda}{4} - \mu mg$ | $\mu = \frac{1}{4}$ | M1 A1
(b) At dist. $x$, $T - \mu mg = -mx$ | $\frac{mg}{4}\left(\frac{1}{4}+x\right) - \frac{1}{4}mg = -mx$ | M1 A1 A1
$x = -\frac{x}{x}$ | S.H.M. with $\omega^2 = \frac{k}{l}$ | A1
(c) Amplitude = $\frac{3l}{4}$ | $x = \frac{3l}{4}\cos\omega t$ | $x = -\frac{l}{4}$; $t = \frac{1}{\omega}\arccos\left(-\frac{1}{3}\right)$ | M1 A1 M1
$= \frac{l}{g}\left(\frac{\pi}{2} + \arcsin\left(\frac{1}{1}\right)\right)s$ | M1 A1
(d) At nat. length, $v^2 = \frac{l}{t}\left(\frac{9l}{16} - \frac{l}{16}\right) = \frac{gl}{2}$ | At $O$, $v^2 = \frac{gl}{2} - 2\frac{l}{k}$, $v=0$ | M1 A1 M1 A1
**Total: 15 marks**A particle $P$ of mass $m$ kg is fixed to one end of a light elastic string of modulus $mg$ N and natural length $l$ m. The other end of the string is attached to a fixed point $O$ on a rough horizontal table. Initially $P$ is at rest in limiting equilibrium on the table at the point $X$ where $OX = \frac{5l}{4}$ m.
\begin{enumerate}[label=(\alph*)]
\item Find the coefficient of friction between $P$ and the table. [2 marks]
\end{enumerate}
$P$ is now given a small displacement $x$ m horizontally along $OX$, away from $O$. While $P$ is in motion, the frictional resistance remains constant at its limiting value.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that as long as the string remains taut, $P$ performs simple harmonic motion with $X$ as the centre. [4 marks]
\end{enumerate}
If $P$ is held at the point where the extension in the string is $l$ m and then released,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item show that the string becomes slack after a time $\left(\frac{\pi}{2} + \arcsin\left(\frac{1}{3}\right)\right)\sqrt{\frac{l}{g}}$ s. [5 marks]
\item Determine the speed of $P$ when it reaches $O$. [4 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q7 [15]}}