The end \(A\) of a uniform rod \(A B\), of length \(2 a\) and weight \(W\), is freely hinged to a vertical wall. The end \(B\) of the rod is attached to a light elastic string of natural length \(\frac { 3 } { 2 } a\) and modulus of elasticity \(3 W\). The other end of the string is attached to the point \(C\) on the wall, where \(C\) is vertically above \(A\) and \(A C = 2 a\). A particle of weight \(2 W\) is attached to the rod at the point \(D\), where \(D B = \frac { 1 } { 2 } a\). The angle \(A B C\) is equal to \(\theta\) (see diagram). Show that \(\cos \theta = \frac { 3 } { 4 }\) and find the tension in the string in terms of \(W\). Find the magnitude of the reaction force at the hinge.