2- A simple model for a tracked gantry conveyor system in a factory is shown in Figure. (75 POINTS) k2 Frictionless Bearings ww k₁ m Frictionless Rollers No-Slip Belt and Pulleys The carriage of mass (m) moves on a frictionless track. The pulley is supported on frictionless bearings, and its axis of rotation is fixed. Its moment of inertia about this axis is J. The motion of the carriage is restrained by a spring of stiffness k, as shown. The belt segment that drives the carriage runs over the pulley without slip, and is attached at the other end to a fixed spring of stiffness K. The displacement of the mass is denoted by x and the corresponding rotation of the pulley is denoted by e. When x-0 (and 80), the springs k and k, have an extension of x and xx, respectively, from their unstretched (free) configurations. Assume that the springs will remain in tension throughout the motion of the system. a. Using Newton's second law, first principles, and free-body diagrams, develop an equivalent equation of motion for this system in terms of the response variable x. What is the equivalent mass, and what is the equivalent stiffness of the system? b. Verify the result in part (a) using the energy method.