A mechanical system consists of a mass M, a damper B, two springs and an ideal gearbox (i.e. negligible inertia and friction) as shown in Figure 1. An external force f(t) is exerted on the mass. For the gearbox, an input angular displacement of produces an output angular displacement of 0o =20. At rest with no external force applied, 0,0 = 0 rad. The input and output levers ('arms') connected rigidly to the input and output shafts of the gearbox, respectively, are both of unit length and have negligible mass. Both springs, one (with Hooke's constant K) connected to M and the other (with Hooke's constant K) connected to the output lever of the gearbox, are rigidly attached to the wall on the left. Both the wall and gearbox are inertially stationary. Assume the applied force to be always sufficiently small to ensure that the displacement 12 remains very small. Output & Input Levers K www ee Gearbox I B Input & Output Shafts (Aligned) y M y f(t) Figure 1: Mechanical assembly. a. Draw all the free-body diagrams and then derive the system's equations. (Motivate all steps performed.) b. Draw the state diagram of the system. (Avoid using differentiators.) (8 marks) (12 marks)