For the inverted-pendulum system shown below, assume that the mass of the inverted pendulum is M/4  and evenly distributed along the length of the rod. Also assume that the center of gravity of the pendulum is located at the center of the rod. If θ is small,

a) Derive mathematical models of the systems in the forms of differential equations

b) Derive related transfer functions

c) Determine state-space equations

Y2 -Y1 - yi Y1 M F(t)