6. \begin{figure}[h] \end{figure} Figure 3 shows a block \(A\) with mass \(4 m\) and a block \(B\) with mass \(5 m\).
Block \(A\) is at rest on a rough plane inclined at an angle \(\alpha\) to the horizontal.
Block \(B\) is at rest on a rough plane inclined at an angle \(\beta\) to the horizontal.
The blocks are connected by a light inextensible string which passes over a small smooth pulley at the top of each plane. A small smooth ring \(C\), of mass \(8 m\), is threaded on the string between the pulleys so that \(A , B\) and \(C\) all lie in the same vertical plane. The part of the string between \(A\) and its pulley lies along a line of greatest slope of the plane of angle \(\alpha\). The part of the string between \(B\) and its pulley lies along a line of greatest slope of the plane of angle \(\beta\). The angle between the vertical and the string between each pulley and the ring \(C\) is \(\gamma\).
The two blocks, \(A\) and \(B\), are modelled as particles.
Given that

• \(\tan \alpha = \frac { 5 } { 12 }\) and \(\tan \beta = \frac { 7 } { 24 }\) and \(\tan \gamma = \frac { 3 } { 4 }\)
• the coefficient of friction, \(\mu\), is the same between each block and its plane
• one of the blocks is on the point of sliding up its plane
• the tension in the string is \(T\)
  1. determine, in terms of \(m\) and \(g\), an expression for \(T\),
  2. draw a diagram showing the forces on block \(A\), clearly labelling each of the forces acting on the block,
  3. determine the value of \(\mu\), giving a justification for your answer.
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