Chapter 3, Problem 3.46P

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-foot-long nylon tethers (Figure P3.46a). 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, just like the pendulum analyzed in Problem 3.42 (see Figure P3.42). However, problem 3.42 neglected the mass of the base, but the descent stage also has significant mass. So a better model is the one shown in Figure P3.46b. The rover mass is $m_r$ the descent stage mass is $m_d$ , and the net horizontal component of the thruster forces is f. Among other simplifications, this model does not take into account vertical motion and any rotational motion.

Derive the equations of motion of the system in terms of the angle $\theta$ and the displacement x, with the force f as the given input.