Simple Nature - Light and Matter

her rate of rotation. That is, by pulling her arms in, she substantiallyreduces the time for each rotation.Earth’s slowing rotation and the receding moon example 2The earth’s rotation is actually slowing down very gradually, withthe kinetic energy being dissipated as heat by friction between theland and the tidal bulges raised in the seas by the earth’s gravity.Does this mean that angular momentum is not really perfectlyconserved? No, it just means that the earth is not quite a closedsystem by itself. If we consider the earth and moon as a system,then the angular momentum lost by the earth must be gained bythe moon somehow. In fact very precise measurements of thedistance between the earth and the moon have been carried outby bouncing laser beams off of a mirror left there by astronauts,and these measurements show that the moon is receding fromthe earth at a rate of 4 centimeters per year! The moon’s greatervalue of r means that it has a greater angular momentum, andthe increase turns out to be exactly the amount lost by the earth.In the days of the dinosaurs, the days were significantly shorter,and the moon was closer and appeared bigger in the sky.But what force is causing the moon to speed up, drawing it outinto a larger orbit? It is the gravitational forces of the earth’s tidalbulges. In figure g, the earth’s rotation is counterclockwise (arrow).The moon’s gravity creates a bulge on the side near it, becauseits gravitational pull is stronger there, and an “anti-bulge”on the far side, since its gravity there is weaker. For simplicity, let’sfocus on the tidal bulge closer to the moon. Its frictional force istrying to slow down the earth’s rotation, so its force on the earth’ssolid crust is toward the bottom of the figure. By Newton’s thirdlaw, the crust must thus make a force on the bulge which is towardthe top of the figure. This causes the bulge to be pulledforward at a slight angle, and the bulge’s gravity therefore pullsthe moon forward, accelerating its orbital motion about the earthand flinging it outward.The result would obviously be extremely difficult to calculate directly,and this is one of those situations where a conservationlaw allows us to make precise quantitative statements about theoutcome of a process when the calculation of the process itselfwould be prohibitively complex.g / A view of the earth-moonsystem from above the northpole. All distances have beenhighly distorted for legibility.Restriction to rotation in a planeIs angular momentum a vector, or a scalar? It does have a directionin space, but it’s a direction of rotation, not a straight-linedirection like the directions of vectors such as velocity or force. Itturns out that there is a way of defining angular momentum as avector, but in this section the examples will be confined to a singleplane of rotation, i.e., effectively two-dimensional situations. In thisSection 4.1 Angular Momentum In Two Dimensions 249