Figure 1: Pendulum on a moving cart.
a) (10 points) For the system shown in Figure 1, obtain the equations of motion in the form of a dynamical model: θ(t) = f(θ, x, F), which is a function of the state x(t).
b) (10 points) Linearize the nonlinear model, obtained in (1), around the equilibrium point where the pendulum is vertical to the cart (θ = 0) and present the linear system in state-space model as x(t) = Ax(t) + BF(t).