| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Potential energy with inextensible strings or gravity only |
| Difficulty | Challenging +1.2 This is a multi-part mechanics problem requiring geometric reasoning (Pythagoras), potential energy formulation, calculus for equilibrium (dV/dx = 0), and second derivative test for stability. While it involves several steps and the constraint geometry, each component is a standard M4 technique with clear structure. The algebraic manipulation is moderate but straightforward, making it above average but not exceptionally challenging for Further Maths students. |
| Spec | 6.02e Calculate KE and PE: using formulae6.04e Rigid body equilibrium: coplanar forces |
\includegraphics{figure_3}
A small smooth peg $P$ is fixed at a distance $d$ from a fixed smooth vertical wire. A particle of mass $3m$ is attached to one end of a light inextensible string which passes over $P$. The particle hangs vertically below $P$. The other end of the string is attached to a small ring $R$ of mass $m$, which is threaded on the wire, as shown in Figure 3.
\begin{enumerate}[label=(\alph*)]
\item Show that when $R$ is at a distance $x$ below the level of $P$ the potential energy of the system is
$$3mg \sqrt{(x^2 + d^2)} - mgx + \text{constant}$$
[4]
\item Hence find $x$, in terms of $d$, when the system is in equilibrium.
[3]
\item Determine the stability of the position of equilibrium.
[3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 2013 Q4 [10]}}