A heavy pole, of mass M and length L, is freely hinged to a wall at the point 0. A rope connects the other end of the pole, B, to a fixed point A on the wall above 0. The system is in equilibrium, with the pole making an angle θ with the horizontal, and the rope making an angle α with the horizontal.

Model the pole as a model rod, and the rope as a model string. Take 0 as origin, with the unit vector i horizontal and the unit vector j vertical, as shown in the diagram.

(i) Draw a force diagram showing all the forces acting on the pole.

(ii) Write down an expression for the weight W of the rod in terms of i and j, and show that the tension T in the rope is given by

T = -|T| cos(α)i + |T| sin(α)j.

(iii) Write down the position vectors of the point B and the midpoint of OB.

(iv) Show that the magnitude of the tension in the rope is Mg cosθ/2 sin(α + θ).

(Hint: You will find it helpful to calculate torques about 0.)