\includegraphics{figure_6}

A particle $P$ of mass $m$ kg is attached to one end of a light inextensible string of length $L$ m. The other end of the string is attached to a fixed point $O$. The particle is held at rest with the string taut and then released. $P$ starts to move and in the subsequent motion the angular displacement of $OP$, at time $t$ s, is $\theta$ radians from the downward vertical (see diagram). The initial value of $\theta$ is $0.05$.

\begin{enumerate}[label=(\roman*)]
\item Show that $\frac{d^2\theta}{dt^2} = -\frac{g}{L} \sin \theta$. [2]

\item Hence show that the motion of $P$ is approximately simple harmonic. [2]

\item Given that the period of the approximate simple harmonic motion is $\frac{4}{3}\pi$ s, find the value of $L$. [2]

\item Find the value of $\theta$ when $t = 0.7$ s, and the value of $t$ when $\theta$ next takes this value. [4]

\item Find the speed of $P$ when $t = 0.7$ s. [3]
\end{enumerate}

\hfill \mbox{\textit{OCR M3 2009 Q6 [13]}}