A trailer of mass \(M\) is connected to a wall through a spring of stiffness \(k_{1}\) and can move on a frictionless horizontal surface, as shown in Fig. 5.50. A uniform cylinder of mass \(m\), connected to the wall of the trailer by a spring of stiffness \(k_{2}\), can roll on the floor of the trailer without slipping. Derive the equations of motion of the system and discuss the nature of coupling present in the system.