| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Session | Specimen |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Potential energy with elastic strings/springs |
| Difficulty | Challenging +1.8 This M4 question requires setting up potential energy for a rod-string system with elastic PE and gravitational PE, then using calculus to find equilibrium. While it involves multiple energy components and geometric reasoning with the constraint that B is below R, the approach is systematic: express extensions/heights in terms of θ, differentiate for equilibrium, and use second derivative test for stability. The 'show that' in part (a) provides scaffolding. More demanding than typical M1/M2 but standard for M4 energy methods. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.04e Rigid body equilibrium: coplanar forces |
\includegraphics{figure_3}
A uniform rod $AB$ has mass $m$ and length $2a$. The end $A$ is smoothly hinged at a fixed point on a fixed straight horizontal wire. A smooth light ring $R$ is threaded on the wire. The ring $R$ is attached by a light elastic string, of natural length $a$ and modulus of elasticity $mg$, to the end $B$ of the rod. The end $B$ is always vertically below $R$ and angle $\angle RAB = \theta$, as shown in Fig. 3.
\begin{enumerate}[label=(\alph*)]
\item Show that the potential energy of the system is
$$mga(2\sin^2\theta - 3\sin\theta) + \text{constant}.$$
[6]
\item Hence determine the value of $\theta$, $0 < \frac{\pi}{2}$, for which the system is in equilibrium.
[5]
\item Determine whether this position of equilibrium is stable or unstable.
[5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 Q7 [16]}}