| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2006 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Potential energy with elastic strings/springs |
| Difficulty | Challenging +1.2 This is a multi-step M4 mechanics problem requiring setup of a geometric configuration, calculation of elastic and gravitational potential energy, and equilibrium analysis using calculus. While it involves several components (geometry, elastic PE, gravitational PE, differentiation), each step follows standard M4 techniques without requiring novel insight. The constraint that Q is vertically below R simplifies the geometry considerably. Part (b) requires finding equilibrium conditions via dPE/dθ = 0 and analyzing the resulting inequality, which is routine for M4 students. More challenging than average due to the algebraic manipulation and multi-component setup, but still a standard examination question for this advanced module. |
| Spec | 6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle6.04e Rigid body equilibrium: coplanar forces |
\includegraphics{figure_1}
A uniform rod $PQ$ has mass $m$ and length $2l$. A small smooth light ring is fixed to the end $P$ of the rod. This ring is threaded on to a fixed horizontal smooth straight wire. A second small smooth light ring $R$ is threaded on to the wire and is attached by a light elastic string, of natural length $l$ and modulus of elasticity $kmg$, to the end $Q$ of the rod, where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that, when the rod $PQ$ makes an angle $\theta$ with the vertical, where $0 < \theta \leq \frac{\pi}{3}$, and $Q$ is vertically below $R$, as shown in Figure 1, the potential energy of the system is
$$mgl[2k\cos^2\theta - (2k + 1)\cos\theta] + \text{constant}.$$
[7]
\end{enumerate}
Given that there is a position of equilibrium with $\theta > 0$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item show that $k > \frac{1}{2}$.
[5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 2006 Q4 [12]}}