9.
\begin{figure}[h]
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\caption{Figure 3}
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A plank, \(A B\), of mass \(M\) and length \(2 a\), rests with its end \(A\) against a rough vertical wall. The plank is held in a horizontal position by a rope. One end of the rope is attached to the plank at \(B\) and the other end is attached to the wall at the point \(C\), which is vertically above \(A\).
A small block of mass \(3 M\) is placed on the plank at the point \(P\), where \(A P = x\). The plank is in equilibrium in a vertical plane which is perpendicular to the wall.
The angle between the rope and the plank is \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Figure 3 .
The plank is modelled as a uniform rod, the block is modelled as a particle and the rope is modelled as a light inextensible string.
- Using the model, show that the tension in the rope is \(\frac { 5 M g ( 3 x + a ) } { 6 a }\)
The magnitude of the horizontal component of the force exerted on the plank at \(A\) by the wall is \(2 M g\).
- Find \(x\) in terms of \(a\).
The force exerted on the plank at \(A\) by the wall acts in a direction which makes an angle \(\beta\) with the horizontal.
- Find the value of \(\tan \beta\)
The rope will break if the tension in it exceeds \(5 M g\).
- Explain how this will restrict the possible positions of \(P\). You must justify your answer carefully.