(Solved): 2. Double rack and pinion system. motor provides a torque Ta the small pinion gear, as shown ...

2. Double rack and pinion system. $\Lambda$ motor provides a torque $T_a$ the small pinion gear, as shown right. The pinion gear drives the rack $m_1$, which moves vertically, and also drives a larger gear connected to another rack $m_2$, which also moves vertically. The two racks are connected to the ceiling via a spring, and to a floor via a damper, respectively. Find a differential equation relating the torque $T_a$ to the height $x$ of the first rack. Neglect gravity. Your solution should only contain the quantities labeled in the diagram.