The system consists of a motorised trolley on a see-saw (see attached figure). The trolley motor is to be controlled and eventually the trolley must balance the beam. A potentiometer senses the angle of the beam at the centre, and relates this to the required direction of travel for the trolley.
To re-iterate, full derivation of equations of motion (Newton) and their corresponding transfer functions are needed.
In your problem, a cart moving to the right moves clockwise to the beam at an angular speed equal to:
dO/dt = Oo + X*(M*g)*cos(O)*t/I
Where Oo is the initial angle, X is the position X from center, M is the cart mass, ...
This solution explains in 177 words the various motion equations for a system. Differential equations for angular speed and velocity are provided along with a Simulink model to describe the system. The derivation of equations of motion are given.