The torsional spring at end A of the uniform slender bar is adjusted so that the bar is in equilibrium in the position θ = θ0, where θ0 is not necessarily small. If the rotational stiffness of the spring is k (torque/rad), determine the expression for the natural frequency of the bar for small oscillations about the equilibrium position. Assume that the plane of the figure is
(a) vertical; and
(b) horizontal.
