Book 2 in the Light and Matter series of free introductory physics textbooks www.com The Light and Matter series of introductory physics textbooks: 1 Newtonian Physics 2 Conservation Laws 3 Vibrations and Waves 4 Electricity and Magnetism 5 Optics 6 The Modern Revolution in Physics www.com Benjamin Crowell www.com Fullerton, California www.com copyright 1998-2004 Benjamin Crowell edition 2. 6th 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 To Uri Haber-Schaim, John Dodge, Robert Gardner, and Edward Shore.com Brief Contents 1 Conservation of Energy 13 2 Simplifying the Energy Zoo 35 3 Work: The Transfer of Mechanical Energy 49 4 Conservation of Momentum 75 5 Conservation of Angular Momentum 105 A Thermodynamics 139 www.com Contents 3 Work: The Transfer of Me- chanical Energy 3.1 Work: The Transfer of Mechanical Energy.
49 The concept of work, 49.—Calculating work as force multiplied by distance, 50.— Machines can increase force, but not work.—No work is done without motion.—Positive and negative work, 53.2 Work in Three Dimensions. 56 A force perpendicular to the motion does 1 Conservation of Energy no work.—Forces at other angles, 56.1 The Search for a Perpetual Motion 3.4 Applications of Calculus .5 Work and Potential Energy .3 A Numerical Scale of Energy. 18 How new forms of energy are discovered, 3.6 ? When Does Work Equal Force 20.7 ? The Dot Product. 67 Energy and relative motion, 24.
30 2 Simplifying the Energy Zoo 2.1 Heat is Kinetic Energy .2 Potential Energy: Energy of Distance or Closeness. 38 An equation for gravitational potential energy, 39.3 All Energy is Potential or Kinetic. 45 4 Conservation of Momentum 4. 76 A conserved quantity of motion, 76.—Generalization of the momentum concept, 79.—Momentum compared to kinetic energy, 81.2 Collisions in One Dimension.
83 The discovery of the neutron, 85.3 ? Relationship of Momentum to the 118.—The torque due to gravity, 120. Center of Mass. 123 Momentum in different frames of reference, Equilibrium, 123.—Stable and unstable 89.—The center of mass frame of reference, equilibria, 125.6 Simple Machines: The Lever .7 ? Proof of Kepler’s Elliptical Orbit Law129 The rate of change of momentum, 91. The area under the force-time graph, 93.5 Momentum in Three Dimensions.
133 The center of mass, 94.—Counting equa- tions and unknowns, 95.—Calculations with R the momentum vector, 96.6 Applications of Calculus .1 Pressure and Temperature. 5 Conservation of Angular A.2 Microscopic Description of an Ideal Momentum Gas .1 Conservation of Angular Momentum 107 Evidence for the kinetic theory, 147.— Restriction to rotation in a plane, 111. Pressure, volume, and temperature, 147.2 Angular Momentum in Planetary A. 112 Efficiency and grades of energy, 151.3 Two Theorems About Angular Heat engines, 151.4 Torque: the Rate of Transfer of Angu- lar Momentum.
117 Appendix 1: Exercises 160 Torque distinguished from force, 117.— Appendix 2: Photo Credits 161 Relationship between force and torque, Appendix 3: Hints and Solutions 162 11 www.com In July of 1994, Comet Shoemaker-Levy struck the planet Jupiter, de- positing 7 × 1022 joules of energy, and incidentally giving rise to a series of Hollywood movies in which our own planet is threatened by an impact by a comet or asteroid. There is evidence that such an impact caused the extinction of the dinosaurs. Left: Jupiter’s gravitational force on the near side of the comet was greater than on the far side, and this differ- ence in force tore up the comet into a string of fragments. Two separate telescope images have been combined to create the illusion of a point of view just behind the comet.
(The colored fringes at the edges of Jupiter are artifacts of the imaging system.) Top: A series of images of the plume of superheated gas kicked up by the impact of one of the fragments. The plume is about the size of North America. Bottom: An image after all the impacts were over, showing the damage done. Chapter 1 Conservation of Energy 1.1 The Search for a Perpetual Motion Machine Don’t underestimate greed and laziness as forces for progress.
Mod- ern chemistry was born from the collision of lust for gold with dis- taste for the hard work of finding it and digging it up. Failed efforts by generations of alchemists to turn lead into gold led finally to the conclusion that it could not be done: certain substances, the chem- ical elements, are fundamental, and chemical reactions can neither 13 www.com increase nor decrease the amount of an element such as gold. Now flash forward to the early industrial age. Greed and laziness have created the factory, the train, and the ocean liner, but in each of these is a boiler room where someone gets sweaty shoveling the coal to fuel the steam engine.
Generations of inventors have tried to create a machine, called a perpetual motion machine, that would run forever without fuel. Such a machine is not forbidden by Newton’s laws of motion, which are built around the concepts of force and inertia. Force is free, and can be multiplied indefinitely with pulleys, gears, or levers. The principle of inertia seems even to encourage the belief that a cleverly constructed machine might not ever run down.
Figures a and b show two of the innumerable perpetual motion machines that have been proposed. The reason these two examples don’t work is not much different from the reason all the others have failed. Even if we assume that a properly shaped ramp would keep the ball rolling smoothly through each cycle, friction would always be at work. The designer imagined that the machine would repeat the same motion over and over again, so that every time it reached a given point its speed would be exactly a / The magnet draws the ball to the top of the ramp, where the same as the last time.
But because of friction, the speed would it falls through the hole and rolls actually be reduced a little with each cycle, until finally the ball back to the bottom. would no longer be able to make it over the top. Friction has a way of creeping into all moving systems. The rotating earth might seem like a perfect perpetual motion machine, since it is isolated in the vacuum of outer space with nothing to exert frictional forces on it.
But in fact our planet’s rotation has slowed drastically since it first formed, and the earth continues to slow its rotation, making today just a little longer than yesterday. The very subtle source of friction is the tides. The moon’s gravity raises bulges in the earth’s oceans, and as the earth rotates the bulges progress around the planet. Where the bulges encounter land, there is friction, which slows the earth’s rotation very gradually.2 Energy b / As the wheel spins clock- wise, the flexible arms sweep The analysis based on friction is somewhat superficial, however.
One around and bend and unbend. By could understand friction perfectly well and yet imagine the follow- dropping off its ball on the ramp, ing situation. Astronauts bring back a piece of magnetic ore from the arm is supposed to make the moon which does not behave like ordinary magnets. A normal itself lighter and easier to lift over bar magnet, c/1, attracts a piece of iron essentially directly toward the top.
Picking its own ball back it, and has no left- or right-handedness. The moon rock, however, up again on the right, it helps to exerts forces that form a whirlpool pattern around it, 2. NASA pull the right side down. goes to a machine shop and has the moon rock put in a lathe and machined down to a smooth cylinder, 3.
If we now release a ball bearing on the surface of the cylinder, the magnetic force whips it around and around at ever higher speeds. Of course there is some 14 Chapter 1 Conservation of Energy www.com friction, but there is a net gain in speed with each revolution. Physicists would lay long odds against the discovery of such a moon rock, not just because it breaks the rules that magnets nor- mally obey but because, like the alchemists, they have discovered a very deep and fundamental principle of nature which forbids cer- tain things from happening. The first alchemist who deserved to be called a chemist was the one who realized one day, “In all these attempts to create gold where there was none before, all I’ve been doing is shuffling the same atoms back and forth among different test tubes.
The only way to increase the amount of gold in my lab- oratory is to bring some in through the door.” It was like having some of your money in a checking account and some in a savings ac- count. Transferring money from one account into the other doesn’t change the total amount. We say that the number of grams of gold is a conserved quan- tity. In this context, the word “conserve” does not have its usual meaning of trying not to waste something.
In physics, a conserved quantity is something that you wouldn’t be able to get rid of even if you wanted to. Conservation laws in physics always refer to a closed system, meaning a region of space with boundaries through which the quantity in question is not passing. In our example, the alchemist’s laboratory is a closed system because no gold is coming c / A mysterious moon rock in or out through the doors. makes a perpetual motion Conservation of mass example 1 machine.
In figure d, the stream of water is fatter near the mouth of the faucet, and skinnier lower down. This is because the water speeds up as it falls. If the cross-sectional area of the stream was equal all along its length, then the rate of flow through a lower cross-section would be greater than the rate of flow through a cross-section higher up. Since the flow is steady, the amount of water between the two cross-sections stays constant.
The cross-sectional area of the stream must therefore shrink in inverse proportion to the increasing speed of the falling water. This is an example of conservation of mass. In general, the amount of any particular substance is not con- served. Chemical reactions can change one substance into another, and nuclear reactions can even change one element into another.
The total mass of all substances is however conserved: the law of conservation of mass The total mass of a closed system always remains constant. Energy cannot be created or destroyed, but only transferred from one system to another. A similar lightbulb eventually lit up in the heads of the people who had been frustrated trying to build a perpetual motion machine. In perpetual motion machine a, consider the motion of one of the d / Example 1.
It performs a cycle of rising and falling. On the way down it gains speed, and coming up it slows back down. Having a greater Section 1.com speed is like having more money in your checking account, and being high up is like having more in your savings account. The device is simply shuffling funds back and forth between the two.
Having more balls doesn’t change anything fundamentally. Not only that, but friction is always draining off money into a third “bank account:” heat. The reason we rub our hands together when we’re cold is that kinetic friction heats things up.