| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2014 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Potential energy with inextensible strings or gravity only |
| Difficulty | Challenging +1.8 This M4 question requires setting up potential energy expressions for a rod-pulley-particle system, then using calculus to find equilibrium and test stability. Part (a) involves geometric reasoning to find heights and string length as functions of θ, requiring careful coordinate work. Part (b) demands differentiation of a trigonometric expression, solving a non-trivial equation involving sin(θ/2) and cos(θ/2), and applying the second derivative test. While systematic, it requires strong geometric visualization, multi-step algebraic manipulation, and integration of several mechanics concepts beyond routine textbook exercises. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.04e Rigid body equilibrium: coplanar forces |
\includegraphics{figure_1}
A uniform rod $AB$, of length $2l$ and mass $12m$, has its end $A$ smoothly hinged to a fixed point. One end of a light inextensible string is attached to the other end $B$ of the rod. The string passes over a small smooth pulley which is fixed at the point $C$, where $AC$ is horizontal and $AC = 2l$. A particle of mass $m$ is attached to the other end of the string and the particle hangs vertically below $C$.
The angle $BAC$ is $\theta$, where $0 < \theta < \frac{\pi}{2}$, as shown in Figure 1.
\begin{enumerate}[label=(\alph*)]
\item Show that the potential energy of the system is
$$4mgl\left(\sin\frac{\theta}{2} - 3\sin\theta\right) + \text{constant}$$
(4)
\end{enumerate}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the value of $\theta$ when the system is in equilibrium and determine the stability of this equilibrium position.
(10)
\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 2014 Q5}}