7.
\begin{figure}[h]
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\caption{Figure 2}
\end{figure}
Figure 2 shows two particles \(A\) and \(B\), of masses \(3 m\) and \(4 m\) respectively, attached to the ends of a light inextensible string. Initially \(A\) is held at rest on the surface of a fixed rough inclined plane. 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 light 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. The particle \(B\) hangs freely and is vertically below \(P\). The system is released from rest with the string taut and with \(B\) at a height of 1.75 m above the ground. In the subsequent motion, \(A\) does not hit the pulley.
For the period before \(B\) hits the ground,
- write down an equation of motion for each particle.
- Hence show that the acceleration of \(B\) is \(\frac { 8 } { 35 } \mathrm {~g}\).
- Explain how you have used the fact that the string is inextensible in your calculation.
When \(B\) hits the ground, \(B\) does not rebound and comes immediately to rest.
- Find the distance travelled by \(A\) from the instant when the system is released to the instant when \(A\) first comes to rest.