$(15$ pts$)$ Consider the inverted pendulum system shown in the figure.

a$.$ Assuming $(t)=0$ when the pendulum is vertically upright, develop a free$-$body diagram of

this system.

b$.$ Using the free$-$body diagram from part $($a$),$ derive the differential equations of motion for the

system. Organizing this expression so that all terms that contain $(t)$ or its derivatives are on

the left$-$hand side of the equation, apply the small$-$angle approximation and find the

equivalent stiffness $K_{eq}$ :

c$.$ Is the equivalent stiffness Keq from part $($b$)$ positive or negative? What might this mean for

the behavior of the system?