1. Consider an inverted pendulum in the gravitv eld; see Fig. l. The length of the pendulum is l and the mass of the pendulum is m. 111 addition. the pendulum is loaded by a torsional spring and a torsional viseous damper at the hinged support. The nnstretehed state of the torsional spring is at the upright position, and the torque generated by the spring is given by T : k9. where it is the spring constant. Similarly, the torque produced by the viscous damper is T 2 ed, where e is the eoeilieient of viscous damping. {a} Find the governing equation of the inverted pendulum. {h} Show that there is only one equilibrium state when k: r: rrtgt, and determine the equi librium state. Linearize the equation of motion and derive the equation governing small motion around the equilibrium state. (e) Consider the ease when k?