A torsion-bar spring consists of a prismatic bar, usually of round cross section, that is twisted at one end and held fast at the other to form a stiff spring. An engineer needs a stiffer one than usual and so considers building in both ends and applying the torque somewhere in the central portion of the span, as shown in the figure. If the bar is uniform in diameter, that is, if d = d1 = d2, investigate how the allowable angle of twist, the largest torque, and the spring rate depend on the location x at which the torque is applied. Hint: Consider two springs in parallel.