4 A uniform beam AB of mass 6 kg and length 5 m rests with its end A on smooth horizontal ground and its end B against a smooth vertical wall. The vertical distance between the ground and B is 4 m , and the angle between the beam and the downward vertical is \(\theta\). To prevent the beam from sliding, one end of a light taut rope of length 2 m is attached to the beam at C and the other end of the rope is attached to a point on the wall 2 m above the ground, as shown in the diagram.
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1. By considering the value of \(\cos \theta\), determine the distance BC . An object of mass 75 kg is placed on the beam at a point which is \(x \mathrm {~m}\) from A . It is given that the tension in the rope is \(T \mathrm {~N}\) and the magnitude of the normal contact force between the ground and the beam is \(R \mathrm {~N}\).
2. By taking moments about B for the beam, show that \(25 \mathrm { R } + 3675 \mathrm { x } - 16 \mathrm {~T} = 19110\).
3. Given that the rope can withstand a maximum tension of 720 N , determine the largest possible value of \(x\).