Consider the simplified system of Fig. 2 , with a cart of negligible mass. Derive the differential equation governing θ.

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Consider the simplified system of Fig. 2 , with a cart of negligible mass. Derive the differential equation governing θ. For this purpose, just represent the force as F(t). (That is, don't assume anything yet about a control scheme.) HINT #1: The sum of the moments about the mass center is equal to the moment of inertia obout the mass center of the pendulum times the angular acceleration. HINT #2: There is an easy formula for the moment of inertia of a point mass about a given axis. HINT #3: The center of mass of the pendulum is probobly where all of the mass of the pendulum is θ. Figure 2 . Schematic of an inverted pendulum attached to a cart of negligible mass. Linearize the differential equation by assuming that θ is small. (The result is called the small angle approximation.) For small θ it can be assumed that the vertical acceleration of the ball is negligible. This latter assumption enables you to determine the (constant) value of the vertical reaction force that is exerted by the cart on the rod. Assume a proportional-derivative (PD) control scheme (i.e., no integral component). Substitute the expression for a PD control scheme for F(t) in the differential equation derived under Task 2 . Then write the differential equation with all terms placed on the lefthand side. (That is, the righthand side should be zero.)