7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6e887c0f-911a-4f18-81a1-2f90025f5410-12_337_1084_228_429}
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\caption{Figure 5}
\end{figure}
Figure 5 shows two particles \(A\) and \(B\), of mass \(2 m\) and \(4 m\) respectively, connected by a light inextensible string. Initially \(A\) is held at rest on a rough inclined plane which is fixed to horizontal ground. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 4 }\). The string passes over a small smooth pulley \(P\) which is fixed at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane and \(B\) hangs vertically below \(P\). The system is released from rest with the string taut, with \(A\) at the point \(X\) and with \(B\) at a height \(h\) above the ground.
For the motion until \(B\) hits the ground,
- give a reason why the magnitudes of the accelerations of the two particles are the same,
- write down an equation of motion for each particle,
- find the acceleration of each particle.
Particle \(B\) does not rebound when it hits the ground and \(A\) continues moving up the plane towards \(P\). Given that \(A\) comes to rest at the point \(Y\), without reaching \(P\),
- find the distance \(X Y\) in terms of \(h\).