(Solved): (a) Obtain a mathematical model for the system shown in the following figure. (Use the Newton's se ...

(a) Obtain a mathematical model for the system shown in the following figure. (Use the Newton's second law) (b) The following figure shows a homogeneous cylinder of radius \( R \) and mass \( m \) that is free to rotate about its axis of rotation and that is connected to the wall through a spring. Assuming that the cylinder rolls on a rough surface without sliding, and \( x \) and \( \theta \) are measured from respective equilibrium positions, derive the equations of motion. Note that the moment of inertia \( J \) for the cylinder is equal to \( \frac{1}{2} m R^{2} \). (Use the law of conservation of energy)