QU$.2(10$ Marks$)$: For the cart and pendulum shown below, the cart must be moved so that mass $m$ is always

in an upright position. The state variables must be expressed in terms of the angular rotation $(t)$ and the

position of the cart $y(t).$ The differential equations describing the motion of the system can be obtained by

writing the sum of the forces in the horizontal direction and the sum of the moments about the pivot point. We

will assume that $Mm$ and the angle of rotation $(t),$ is small so that the equations are linear.