The "sky crane" shown on the text cover was a novel solution to the problem of landing the 2000 lb Curiosity rover on the surface of Mars. Curiosity hangs from the descent stage by 60-ft long nylon tethers (Figure a). The descent stage uses its thrusters to hover as the rover is lowered to the surface. Thus the rover behaves like a pendulum whose base is moving horizontally. The side thruster force is not constant but is controlled to keep the descent stage from deviating left or right from the desired vertical path. As we will see in Chapter 10, such a control system effectively acts like a spring and a damper, as shown in Figure b. The rover mass is mr. the descent stage mass is md, and the net horizontal component of the thruster forces is f = kx + e. Among other simplifications, this model neglects vertical motion and any rotational motion. Derive the equations of motion of the system in terms of the angle θ and the displacement x.
a.


b.