Consider a simple pendulum, consisting of a rod of length 7, and assume that all the mass m is lumped at the end of the pendulum. There is an external torque t applied at the pivot, acting as the input, and a torque proportional to the derivative of the angular velocity with coefficient c, due to the resistance of the air. T l mg a) Figure P1 Sketch the free body diagram. Hence write the single governing equation which describes in terms of the states and input of the system, defining the state vector and the input vector respectively as, x(t) = [ 0 ] and u(t) = T(t). b) Verify that there are two equilibria that corresponds to = 0: one for 0 = 0 (pendulum down) and one for = (pendulum up). Linearize the non-linear equation x(t) = f[x,(t), u(t)] around the equilibrium that corresponds to 0 = 0. c) Are there equilibria that correspond to 0? Which ones?