Simple Nature - Light and Matter

y a gas or liquid, like the turbine in a hydroelectric power plant.Figure d shows two types of surfaces that could be attached tothe circumference of an old-fashioned waterwheel. Compare theforce exerted by the water in the two cases.⊲ Let the x axis point to the right, and the y axis up. In bothcases, the stream of water rushes down onto the surface withmomentum p y,i = −p o , where the subscript i stands for “initial,”i.e., before the collision.In the case of surface 1, the streams of water leaving the surfacehave no momentum in the y direction, and their momenta in the xdirection cancel. The final momentum of the water is zero alongboth axes, so its entire momentum, −p o , has been transferred tothe waterwheel.When the water leaves surface 2, however, its momentum isn’tzero. If we assume there is no friction, it’s p y,f = +p o , with thepositive sign indicating upward momentum. The change in thewater’s momentum is p y,f − p y,i = 2p o , and the momentum transferredto the waterwheel is −2p o .Force is defined as the rate of transfer of momentum, so surface2 experiences double the force. A waterwheel constructed in thisway is known as a Pelton waterwheel.The Yarkovsky effect example 53We think of the planets and asteroids as inhabiting their orbitspermanently, but it is possible for an orbit to change over periodsof millions or billions of years, due to a variety of effects. Forasteroids with diameters of a few meters or less, an importantmechanism is the Yarkovsky effect, which is easiest to understandif we consider an asteroid spinning about an axis that is exactlyperpendicular to its orbital plane.The illuminated side of the asteroid is relatively hot, and radiatesmore infrared light than the dark (night) side. Light has momentum,and a total force away from the sun is produced by combinedeffect of the sunlight hitting the asteroid and the imbalance betweenthe momentum radiated away on the two sides. This force,however, doesn’t cause the asteroid’s orbit to change over time,since it simply cancels a tiny fraction of the sun’s gravitationalattraction. The result is merely a tiny, undetectable violation ofKepler’s law of periods.Consider the sideways momentum transfers, however. In figure e,the part of the asteroid on the right has been illuminated for half aspin-period (half a “day”) by the sun, and is hot. It radiates morelight than the morning side on the left. This imbalance producesa total force in the x direction which points to the left. If the asteroid’sorbital motion is to the left, then this is a force in the samedirection as the motion, which will do positive work, increasingd / Two surfaces that couldbe used to extract energy from astream of water.e / An asteroid absorbs visiblelight from the sun, and getsrid of the energy by radiatinginfrared light.Section 3.4 Motion In Three Dimensions 189