A spacecraft design similar to GEOSAT proposes to use passive gravity-gradient stabilization for pitch attitude control (motion about the y-axis). The satellite consists of a tip mass, separated from the main satellite body mass by a rigid boom with length 10 m. The z-body axis is along the boom, and pitching motion is along the y-body axis (which is opposite the angular momentum vector of the orbit; the +y-axis is into the page). The spacecraft is in circular LEO (altitude = 800 km). The satellite is axis-symmetric, and the moments of inertia are Izz = 85 kg-m2, and Ixx = Iyy =2530 kg-m2.
Compute (by hand) the pitching motion of the spacecraft in LEO for an initial pitch angle of 5 deg. Assume that the only torque acting on the satellite is the gravity-gradient torque and that the initial angular rates are zero and that no roll/yaw motion exists. Sketch the pitching motion vs. time for one orbital revolution.