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 I_zz = 85 kg-m², and  I_xx = I_yy =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.