a) State Castiglianos theorem for translational and rotational displacements of an elastic body, stating precisely the meanings of the terms.

b) A swing in a childrens play area is constrcuted from a steel tube bent into a quarter circle of radius R. One end is rigidly fixed to the ground with the tangent to the circle vertical, and the swing attached to the other end. Assuming that the beam has a section constant EI, derive experssions for the vertical and horizontal displacements of the swing when a downwards load P is applied to it.