7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a1365c54-4910-449b-b270-c56c1bc5a751-24_396_992_246_539}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity 2 mg . The other end of the string is attached to a fixed point \(O\) on a rough plane which is inclined to the horizontal at an angle \(\alpha\)
The string lies along a line of greatest slope of the plane.
The particle \(P\) is held at rest on the plane at the point \(A\), where \(O A = a\), as shown in Figure 5.
The particle \(P\) is released from \(A\) and slides down the plane, coming to rest at the point \(B\). The coefficient of friction between \(P\) and the plane is \(\mu\), where \(\mu < \tan \alpha\) Air resistance is modelled as being negligible.
- Show that \(A B = a ( \sin \alpha - \mu \cos \alpha )\).
Given that \(\tan \alpha = \frac { 3 } { 4 }\) and \(\mu = \frac { 1 } { 2 }\)
- find, in terms of \(a\) and \(g\), the maximum speed of \(P\) as it moves from \(A\) to \(B\)
- Describe the motion of \(P\) after it reaches \(B\), justifying your answer.