copyright 2006 Benjamin Crowell rev. 14th October 2006 This book is licensed under the Creative Com- mons Attribution-ShareAlike license, version 1.org/licenses/by-sa/1.0/, except for those photographs and drawings of which I am not the author, as listed in the photo credits. If you agree to the license, it grants you certain privileges that you would not otherwise have, such as the right to copy the book, or download the digital version free of charge from www. At your option, you may also copy this book under the GNU Free Documentation License version 1.org/licenses/fdl.txt, with no invariant sections, no front-cover texts, and no back-cover texts.com Brief Contents 1 Conservation of Mass and Energy 7 2 Conservation of Momentum 37 3 Conservation of Angular Momentum 61 4 Relativity 69 5 Electricity 91 6 Fields 109 7 The Ray Model of Light 127 8 Waves 155 For a semester-length course, all seven chapters can be covered.
For a shorter course, the book is designed so that chapters 1, 2, and 5 are the only ones that are required for conti- nuity; any of the others can be included or omitted at the instructor’s discretion, with the only constraint being that chapter 6 requires chapter 4.com Contents Momentum compared to kinetic energy, 45.—Motion in two dimensions, 49. 58 3 Conservation of Angular Momentum 3. 61 1 Conservation of Mass and 3. 65 Energy Torque distinguished from force, 66.3 Noether’s Theorem for Angular 1.1 Symmetry and Conservation Laws .2 Conservation of Mass .3 Review of the Metric System and Conversions.
11 The Metric System, 11.4 Conservation of Energy .—The principle of inertia, 16.—Energy in general, 21.5 Newton’s Law of Gravity .6 Noether’s Theorem for Energy.7 Equivalence of Mass and Energy .—The correspondence principle, 30.1 The Principle of Relativity.2 Distortion of Time and Space. 82 Combination of velocities, 83.— 2 Conservation of Momentum Momentum, 84.—Equivalence of mass and energy, 87.2 The Principle of Inertia. 39 Symmetry and inertia, 39. 40 5 Electricity Conservation of momentum, 40.1 The Quest for the Atomic Force .2 Charge, Electricity and Magnetism.
93 7 The Ray Model of Light Charge, 93.—Conservation of charge, 95. 127 Electrical forces involving neutral objects, The nature of light, 128.—The atom, and subatomic particles, of light with matter, 131. model of light, 132.—Geometry of specu- 5. 136 The volt unit, 99.
The inverse-square law, 136.3 ? The Principle of Least Time for Problems .4 Images by Reflection .—Images of images, 146.1 Farewell to the Mechanical Universe 109 Time delays in forces exerted at a distance, 110.—More evidence that fields of force are 8 Waves real: they carry energy.—Sources and sinks, 8.—The electric field, 113. 113 is not transported with the wave.—Relativity re- A wave’s velocity depends on the medium.3 Sound and Light Waves. 125 Period and frequency of a periodic wave, 165.—Graphs of waves as a function of position, 165.— Wave velocity related to frequency and wavelength, 166. 169 Appendix 1: Photo Credits 171 Appendix 2: Hints and Solutions 173 5 www.com Chapter 1 Conservation of Mass and Energy 1.1 Symmetry and Conservation Laws Even before history began, people must already have noticed certain facts about the sky.
The sun and moon both rise in the east and set in the west. Another fact that can be settled to a fair degree of accuracy using the naked eye is that the apparent sizes of the sun and moon don’t change noticeably. (There is an optical illusion that makes the moon appear bigger when it’s near the horizon, but you can easily verify that it’s nothing more than an illusion, by checking its angular size against some standard, such as your pinkie held at arm’s length.) If the sun and moon were varying their distances from us, they would appear to get bigger and smaller, and since they don’t appear to change in size, it appears, at least approximately, that they always stay at the same distance from us. From observations like these, the ancients constructed a scientific model, in which the sun and moon traveled around the earth in perfect circles.
Of course, we now know that the earth isn’t the center of the universe, but that doesn’t mean the model wasn’t a / Due to the rotation of the useful. That’s the way science always works. Science never aims earth, everything in the sky to reveal the ultimate reality. Science only tries to make models of appears to spin in circles.
In this reality that have predictive power. time-exposure photograph, each star appears as a streak. Our modern approach to understanding physics revolves around the concepts of symmetry and conservation laws, both of which are demonstrated by this example. The sun and moon were believed to move in circles, and a circle is a very symmetric shape.
If you rotate a circle about its center, like a spinning wheel, it doesn’t change. Therefore, we say that the circle is symmetric with respect to rotation about its center. The ancients thought it was beautiful that the universe seemed to have this type of symmetry built in, and became very attached to the idea. A conservation law is a statement that some number stays the same with the passage of time.
In our example, the distance between the sun and the earth is conserved, and so is the distance between the moon and the earth. (The ancient Greeks were even able to determine that earth-moon distance.com In our example, the symmetry and the conservation law both give the same information. Either statement can be satisfied only by a circular orbit. That isn’t a coincidence.
Physicist Emmy Noether showed on very general mathematical grounds that for physical the- ories of a certain type, every symmetry leads to a corresponding conservation law. Although the precise formulation of Noether’s theorem, and its proof, are too mathematical for this book, we’ll see many examples like this in which the physical content of the theorem is fairly straightforward. The idea of perfect circular orbits seems very beautiful and in- tuitively appealing. It came as a great disappointment, therefore, when the astronomer Johannes Kepler discovered, by the painstak- ing study of precise observations, that orbits such as the moon’s were actually ellipses, not circles.
This is the sort of thing that led the biologist Huxley to say, “The great tragedy of science is the slaying of a beautiful theory by an ugly fact.” The lesson of this story, then, is that symmetries are important and beautiful, but we can’t decide which symmetries are right based only on common sense or aesthetics; their validity can only be determined based on b / Emmy Noether (1882-1935). observations and experiments. The daughter of a prominent As a more modern example, consider the symmetry between German mathematician, she did right and left. For example, we observe that a top spinning clockwise not show any early precocity at mathematics — as a teenager has exactly the same behavior as a top spinning counterclockwise.
she was more interested in music This kind of observation led physicists to believe, for hundreds of and dancing. She received her years, that the laws of physics were perfectly symmetric with respect doctorate in 1907 and rapidly to right and left. The symmetry appealed to physicists’ common built a world-wide reputation, sense. However, experiments by Wu et al.
in 1957 showed that but the University of Göttingen this symmetry was violated in certain types of nuclear reactions. refused to let her teach, and her Physicists were thus forced to change their opinions about what colleague Hilbert had to advertise constituted common sense. her courses in the university’s catalog under his own name. A long controversy ensued, with her opponents asking what the 1.2 Conservation of Mass country’s soldiers would think We intuitively feel that matter shouldn’t appear or disappear out of when they returned home and nowhere: that the amount of matter should be a conserved quan- were expected to learn at the tity.
If that was to happen, then it seems as though atoms would feet of a woman. Allowing her have to be created or destroyed, which doesn’t happen in any phys- on the faculty would also mean letting her vote in the academic ical processes that are familiar from everyday life, such as chemical senate. Said Hilbert, “I do not reactions. On the other hand, I’ve already cautioned you against see that the sex of the candidate believing that a law of physics must be true just because it seems is against her admission as a appealing.
The laws of physics have to be found by experiment, and privatdozent [instructor]. After there seem to be experiments that are exceptions to the conserva- all, the university senate is not tion of matter. A log weighs more than its ashes. Did some matter a bathhouse.” She was finally simply disappear when the log was burned? admitted to the faculty in 1919.
A Jew, Noether fled Germany in The French chemist Antoine-Laurent Lavoisier was the first sci- 1933 and joined the faculty at entist to realize that there were no such exceptions. Lavoisier hy- Bryn Mawr in the U. pothesized that when wood burns, for example, the supposed loss 8 Chapter 1 Conservation of Mass and Energy www.com of weight is actually accounted for by the escaping hot gases that the flames are made of. Before Lavoisier, chemists had almost never weighed their chemicals to quantify the amount of each substance that was undergoing reactions.
They also didn’t completely under- stand that gases were just another state of matter, and hadn’t tried performing reactions in sealed chambers to determine whether gases were being consumed from or released into the air. For this they had at least one practical excuse, which is that if you perform a gas- releasing reaction in a sealed chamber with no room for expansion, you get an explosion! Lavoisier invented a balance that was capable of measuring milligram masses, and figured out how to do reactions in an upside-down bowl in a basin of water, so that the gases could expand by pushing out some of the water. In one crucial experi- ment, Lavoisier heated a red mercury compound, which we would now describe as mercury oxide (HgO), in such a sealed chamber. A gas was produced (Lavoisier later named it “oxygen”), driving c / Portrait of Monsieur Lavoisier out some of the water, and the red compound was transformed into and His Wife, by Jacques-Louis silvery liquid mercury metal.
The crucial point was that the total David, 1788. Lavoisier invented mass of the entire apparatus was exactly the same before and after the concept of conservation of the reaction. Based on many observations of this type, Lavoisier mass. The husband is depicted proposed a general law of nature, that matter is always conserved.
with his scientific apparatus, while in the background on the self-check A left is the portfolio belonging In ordinary speech, we say that you should “conserve” something, be- to Madame Lavoisier, who is cause if you don’t, pretty soon it will all be gone. How is this different thought to have been a student of from the meaning of the term “conservation” in physics?. 173 Although Lavoisier was an honest and energetic public official, he was caught up in the Terror and sentenced to death in 1794. He requested a fifteen-day delay of his execution so that he could com- plete some experiments that he thought might be of value to the Republic.
The judge, Coffinhal, infamously replied that “the state has no need of scientists.” As a scientific experiment, Lavoisier de- cided to try to determine how long his consciousness would continue after he was guillotined, by blinking his eyes for as long as possible. He blinked twelve times after his head was chopped off. Ironically, Judge Coffinhal was himself executed only three months later, falling victim to the same chaos.