4 A uniform smooth pulley can rotate freely about its axis, which is fixed and horizontal. A light elastic string AB is attached to the pulley at the end B . The end A is attached to a fixed point such that the string is vertical and is initially at its natural length with B at the same horizontal level as the axis. In this position a particle P is attached to the highest point of the pulley. This initial position is shown in Fig. 4.1.

The radius of the pulley is $a$, the mass of P is $m$ and the stiffness of the string AB is $\frac { m g } { 10 a }$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{636e1d23-3bbb-469f-8fc9-1f64da865126-3_451_517_607_466}
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\caption{Fig. 4.1}
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\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{636e1d23-3bbb-469f-8fc9-1f64da865126-3_456_451_607_1226}
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\caption{Fig. 4.2}
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(i) Fig. 4.2 shows the system with the pulley rotated through an angle $\theta$ and the string stretched. Write down the extension of the string and hence find the potential energy, $V$, of the system in this position. Show that $\frac { \mathrm { d } V } { \mathrm {~d} \theta } = m g a \left( \frac { 1 } { 10 } \theta - \sin \theta \right)$.\\
(ii) Hence deduce that the system has a position of unstable equilibrium at $\theta = 0$.\\
(iii) Explain how your expression for $V$ relies on smooth contact between the string and the pulley.

Fig. 4.3 shows the graph of the function $\mathrm { f } ( \theta ) = \frac { 1 } { 10 } \theta - \sin \theta$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{636e1d23-3bbb-469f-8fc9-1f64da865126-3_538_1342_1706_404}
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\caption{Fig. 4.3}
\end{center}
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(iv) Use the graph to give rough estimates of three further values of $\theta$ (other than $\theta = 0$ ) which give positions of equilibrium. In each case, state with reasons whether the equilibrium is stable or unstable.\\
(v) Show on a sketch the physical situation corresponding to the least value of $\theta$ you identified in part (iv). On your sketch, mark clearly the positions of P and B .\\
(vi) The equation $\mathrm { f } ( \theta ) = 0$ has another root at $\theta \approx - 2.9$. Explain, with justification, whether this necessarily gives a position of equilibrium.

\hfill \mbox{\textit{OCR MEI M4 2008 Q4 [24]}}