Consider the mechanical system shown in Figure 5.93. The inputs are the force \(f_{1}\) applied to the cart and the force \(f_{2}\) applied at the tip of the rod. The outputs are the displacement \(x\) of the cart and the angular displacement \(\theta\) of the pendulum.

a. Draw the free-body diagram and kinematic diagram, and derive the equations of motion for small angles.

b. Using the differential equation obtained in Part (a), determine the state-space representation.

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FIGURE 5.93 Problem 6. 0000 90 M, L x fi