Simple Nature - Light and Matter

inward toward the hinge will have no angular momentum to giveto the door. After all, there would not even be any way to decidewhether the ball’s rotation was clockwise or counterclockwisein this situation. It is therefore only the component of the blob’svelocity vector perpendicular to the door that should be counted inits angular momentum,L = mv ⊥ r .e / Only the component ofthe velocity vector perpendicularto the line connecting the objectto the axis should be countedinto the definition of angularmomentum.More generally, v ⊥ should be thought of as the component of theobject’s velocity vector that is perpendicular to the line joining theobject to the axis of rotation.We find that this equation agrees with the definition of the originalputty blob as having one unit of angular momentum, and wecan now see that the units of angular momentum are (kg·m/s)·m,i.e., kg·m 2 /s. Summarizing, we haveL = mv ⊥ r [angular momentum of a particle in two dimensions] ,where m is the particle’s mass, v ⊥ is the component of its velocityvector perpendicular to the line joining it to the axis of rotation,and r is its distance from the axis. (Note that r is not necessarilythe radius of a circle.) Positive and negative signs of angular momentumare used to describe opposite directions of rotation. Theangular momentum of a finite-sized object or a system of many objectsis found by dividing it up into many small parts, applying theequation to each part, and adding to find the total amount of angularmomentum. (As implied by the word “particle,” matter isn’tthe only thing that can have angular momentum. Light can alsohave angular momentum, and the above equation would not applyto light.)Conservation of angular momentum has been verified over andover again by experiment, and is now believed to be one of the mostfundamental principles of physics, along with conservation of mass,energy, and momentum.f / A figure skater pulls in herarms so that she can execute aspin more rapidly.A figure skater pulls her arms in. example 1When a figure skater is twirling, there is very little friction betweenher and the ice, so she is essentially a closed system, and herangular momentum is conserved. If she pulls her arms in, she isdecreasing r for all the atoms in her arms. It would violate conservationof angular momentum if she then continued rotating atthe same speed, i.e., taking the same amount of time for eachrevolution, because her arms’ contributions to her angular momentumwould have decreased, and no other part of her wouldhave increased its angular momentum. This is impossible becauseit would violate conservation of angular momentum. If hertotal angular momentum is to remain constant, the decrease in rfor her arms must be compensated for by an overall increase in248 Chapter 4 Conservation of Angular Momentum