Consider a rigid body of arbitrary shape which is attached at its mass center O and subjected to no force other than its weight and the reaction of the support at O. (a) Prove that the angular momentum HO of the body about the fixed point O is constant in magnitude and direction, that the kinetic energy T of the body is constant, and that the projection along HO of the angular velocity ω of the body is constant. (b) Show that the tip of the vector ω describes a curve on a fixed plane in space (called the invariable plane), which is perpendicular to HO and at a distance 2T/HO from O. (c) Show that with respect to a frame of reference attached to the body and coinciding with its principal axes of inertia, the tip of the vector ω appears to describe a curve on an ellipsoid of equation
The ellipsoid (called the Poinsot ellipsoid) is rigidly attached to the body and is of the same shape as the ellipsoid of inertia, but of a different size.