\includegraphics{figure_4}

A light elastic spring has natural length $l$ and modulus of elasticity $4mg$. One end of the spring is attached to a point $A$ on a plane that is inclined to the horizontal at an angle $\alpha$, where $\tan\alpha = \frac{3}{4}$. The other end of the spring is attached to a particle $P$ of mass $m$. The plane is rough and the coefficient of friction between $P$ and the plane is $\frac{1}{4}$. The particle $P$ is held at a point $B$ on the plane where $B$ is below $A$ and $AB = l$, with the spring lying along a line of greatest slope of the plane, as shown in Figure 4. At time $t = 0$, the particle is projected up the plane towards $A$ with speed $\frac{1}{2}\sqrt{gl}$. At time $t$, the compression of the spring is $x$.

\begin{enumerate}[label=(\alph*)]
\item Show that
$$\frac{d^2x}{dt^2} + 4\omega^2x = -g, \text{ where } \omega = \sqrt{\frac{g}{l}}.$$
[6]
\end{enumerate}

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find $x$ in terms of $l$, $\omega$ and $t$.
[7]
\end{enumerate}

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the distance that $P$ travels up the plane before first coming to rest.
[4]
\end{enumerate}

\hfill \mbox{\textit{Edexcel M4 2006 Q7 [17]}}