Simple Nature - Light and Matter

surfaces consists only of the two points at the front and back of thesphere. The angular momentum is confined to one of these points,and can’t change its direction, i.e., its orientation with respect to theprincipal axis system, which is another way of saying that the shoecan’t change its orientation with respect to the angular momentumvector. In the bottom figure, the shoe is rotating about the longestaxis. Now the angular momentum vector is trapped at one of thetwo points on the right or left. In the case of rotation about theaxis with the intermediate moment of inertia element, however, theintersection of the sphere and the ellipsoid is not just a pair of isolatedpoints but the curve shown with the dashed line. The relativeorientation of the shoe and the angular momentum vector can andwill change.One application of the moment of inertia tensor is to video gamesthat simulate car racing or flying airplanes.One more exotic example has to do with nuclear physics. Althoughyou have probably visualized atomic nuclei as nothing morethan featureless points, or perhaps tiny spheres, they are often ellipsoidswith one long axis and two shorter, equal ones. Althougha spinning nucleus normally gets rid of its angular momentum viagamma ray emission within a period of time on the order of picoseconds,it may happen that a deformed nucleus gets into a state inwhich has a large angular momentum is along its long axis, whichis a very stable mode of rotation. Such states can live for secondsor even years! (There is more to the story — this is the topic onwhich I wrote my Ph.D. thesis — but the basic insight applies eventhough the full treatment requires fancy quantum mechanics.)Our analysis has so far assumed that the kinetic energy of rotationenergy can’t be converted into other forms of energy such asheat, sound, or vibration. When this assumption fails, then rotationabout the axis of least moment of inertia becomes unstable,and will eventually convert itself into rotation about the axis whosemoment of inertia is greatest. This happened to the U.S.’s first artificialsatellite, Explorer I, launched in 1958. Note the long, floppyantennas, which tended to dissipate kinetic energy into vibration. Ithad been designed to spin about its minimimum-moment-of-inertiaaxis, but almost immediately, as soon as it was in space, it beganspinning end over end. It was nevertheless able to carry out itsscience mission, which didn’t depend on being able to maintain astable orientation, and it discovered the Van Allen radiation belts.This chapter is summarized on page 950. Notation and terminologyare tabulated on pages 941-942.l / Visualizing surfaces ofconstant energy and angularmomentum in L x -L y -L z space.m / The Explorer I satellite.Section 4.3 Angular Momentum In Three Dimensions 287