Refer to Probs. 18.143 and 18.144. (a) Show that the curve (called polhode) described by the tip of the vector V with respect to a frame of reference coinciding with the principal axes of inertia of the rigid body is defined by the equations
Using the results obtained in Probs. 18.143 and 18.144, show that for an axisymmetrical body attached at its mass center O and under no force other than its weight and the reaction at O, the Poinsot ellipsoid is an ellipsoid of revolution and the space and body cones are both circular and are tangent to each other. Further show that (a) the two cones are tangent externally, and the precession is direct, when I
Prob. 18.143, 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.
and that this curve can, therefore, be obtained by intersecting the Poinsot ellipsoid with the ellipsoid defined by Eq. (2). (b) Further show, assuming Ix y z, that the polhodes obtained for various values of HO have the shapes indicated in the figure. (c) Using the result obtained in part b, show that a rigid body under no force can rotate about a fixed centroidal axis if, and only if, that axis coincides with one of the principal axes of inertia of the body, and that the motion will be stable if the axis of rotation coincides with the major or minor axis of the Poinsot ellipsoid (z or x axis in the figure) and unstable if it coincides with the intermediate axis (y axis).