Simple Nature - Light and Matter

4.3 Angular Momentum In Three DimensionsConservation of angular momentum produces some surprising phenomenawhen extended to three dimensions. Try the following experiment,for example. Take off your shoe, and toss it in to the air,making it spin along its long (toe-to-heel) axis. You should observea nice steady pattern of rotation. The same happens when you spinthe shoe about its shortest (top-to-bottom) axis. But somethingunexpected happens when you spin it about its third (left-to-right)axis, which is intermediate in length between the other two. Insteadof a steady pattern of rotation, you will observe something morecomplicated, with the shoe changing its orientation with respect tothe rotation axis.4.3.1 Rigid-body kinematics in three dimensionsHow do we generalize rigid-body kinematics to three dimensions?When we wanted to generalize the kinematics of a moving particleto three dimensions, we made the numbers r, v, and a into vectorsr, v, and a. This worked because these quantities all obeyed thesame laws of vector addition. For instance, one of the laws of vectoraddition is that, just like addition of numbers, vector addition givesthe same result regardless of the order of the two quantities beingadded. Thus you can step sideways 1 m to the right and thenstep forward 1 m, and the end result is the same as if you steppedforward first and then to the side. In order words, it didn’t matterwhether you took ∆r 1 + ∆r 2 or ∆r 2 + ∆r 1 . In math this is calledthe commutative property of addition.a / Performing the rotations in oneorder gives one result, 3, while reversingthe order gives a differentresult, 5.Angular motion, unfortunately doesn’t have this property, asshown in figure a. Doing a rotation about the x axis and then278 Chapter 4 Conservation of Angular Momentum