"Book 2 in the Light and Matter series of free introductory physics textbooks www.lightandmatter.com
The Light and Matter series of introductory physics textbooks: 1 2 3 4 5 6 Newtonian Physics Conservation Laws Vibrations and Waves Electricity and Magnetism Optics The Modern Revolution in Physics
Benjamin Crowell
www.lightandmatter.com
Fullerton, California www.lightandmatter.com
copyright 1998-2004 Benjamin Crowell edition 2.2 rev. 19th November 2006 This book is licensed under the Creative Commons Attribution-ShareAlike license, version 1.0, http://creativecommons.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.lightandmatter.com. At your option, you may also copy this book under the GNU Free Documentation License version 1.2, http://www.gnu.org/licenses/fdl.txt, with no invariant sections, no front-cover texts, and no back-cover texts. ISBN 0-9704670-2-8
To Uri Haber-Schaim, John Dodge, Robert Gardner, and Edward Shore.
Brief Contents
1 2 3 4 5 A Conservation of Energy 13 Simplifying the Energy Zoo 35 Work: The Transfer of Mechanical Energy 49 Conservation of Momentum 75 Conservation of Angular Momentum 105 Thermodynamics 139
Contents
3 Work: The Transfer of Mechanical 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., 52.—No work is done without motion., 52.—Positive and negative work, 53.
3.2 Work in Three Dimensions . . . .
56
1 Conservation of Energy
1.1 The Search for a Perpetual Machine . . . . . . . . . . . 1.2 Energy . . . . . . . . . 1.3 A Numerical Scale of Energy Motion . . . 13 . . . 14 . . . 18
A force perpendicular to the motion does no work., 56.—Forces at other angles, 56.
How new forms of energy are discovered, 20.
1.4 Kinetic Energy . . . . . . . . .
Energy and relative motion, 24.
23 26 28 30
1.5 Power . . . . . . . . . . . . . Summary . . . . . . . . . . . . . Problems . . . . . . . . . . . . .
3.3 Varying Force . . . . . . . 3.4 Applications of Calculus . . 3.5 Work and Potential Energy . . 3.6 When Does Work Equal Times Distance? . . . . . . . . 3.7 The Dot Product . . . . . . Summary . . . . . . . . . . . Problems . . . . . . . . . . .
.. .. .. Force .. .. .. ..
58 61 62 65 67 68 70
2 Simplifying the Energy Zoo
2.1 Heat is Kinetic Energy . . . . . . 36 2.2 Potential Energy: Energy of Distance or Closeness . . . . . . . . . . . . 38
An equation for gravitational potential energy, 39.
2.3 All Energy is Potential or Kinetic . . Summary . . . . . . . . . . . . . Problems . . . . . . . . . . . . .
42 44 45
4 Conservation of Momentum
4.1 Momentum . . . . . . . . . . .
A conserved quantity of motion, 76.— Momentum, 77.—Generalization of the momentum concept, 79.—Momentum compared to kinetic energy, 81.
76
4.2 Collisions in One Dimension . . .
The discovery of the neutron, 85.
83
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4.3 Relationship of Momentum to the Center of Mass . . . . . . . . . . . 88
Momentum in different frames of reference, 89.—The center of mass frame of reference, 90.
119.—The torque due to gravity, 121.
5.5 Statics. . . . . . . . . . . . . 124
Equilibrium, 124.—Stable and unstable equilibria, 126.
4.4 Momentum Transfer . . . . . . .
The rate of change of momentum, 91.— The area under the force-time graph, 93.
91
5.6 Simple Machines: The Lever . . . 128 5.7 Proof of Kepler’s Elliptical Orbit Law130
*, 131.—*, 131.
4.5 Momentum in Three Dimensions. .
The center of mass, 94.—Counting equations and unknowns, 95.—Calculations with the momentum vector, 96.
94
Summary . . . . . . . . . . . . . 132 Problems . . . . . . . . . . . . . 134
4.6 Applications of Calculus . . . . 98 Summary . . . . . . . . . . . . . 99 Problems . . . . . . . . . . . . . 101
A Thermodynamics
A.1 Pressure and Temperature . . . . 140
5 Conservation Momentum
of
Angular
Pressure, 140.—Temperature, 144.
A.2 Microscopic Description of an Ideal Gas . . . . . . . . . . . . . . . 147
Evidence for the kinetic theory, 147.— Pressure, volume, and temperature, 147.
5.1 Conservation of Angular Momentum 107
Restriction to rotation in a plane, 111.
5.2 Angular Momentum in Planetary Motion . . . . . . . . . . . . . . 112 5.3 Two Theorems About Angular Momentum . . . . . . . . . . . . 114 5.4 Torque: the Rate of Transfer of Angular Momentum . . . . . . . . . . . 118
Torque distinguished from force, 118.— Relationship between force and torque,
A.3 Entropy . . . . . . . . . . . . 151
Efficiency and grades of energy, 151.— Heat engines, 151.—Entropy, 153.
Problems . . . . . . . . . . . . . 157
Appendix 1: Exercises 160 Appendix 2: Photo Credits 161 Appendix 3: Hints and Solutions 162
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12
In July of 1994, Comet Shoemaker-Levy struck the planet Jupiter, depositing 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 difference 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. Modern chemistry was born from the collision of lust for gold with distaste 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 chemical elements, are fundamental, and chemical reactions can neither
13
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. Consider machine a. 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 the same as the last time. But because of friction, the speed would actually be reduced a little with each cycle, until finally the ball 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.
a / The magnet draws the ball to the top of the ramp, where it falls through the hole and rolls back to the bottom.
1.2 Energy
b / As the wheel spins clockwise, the flexible arms sweep around and bend and unbend. By dropping off its ball on the ramp, the arm is supposed to make itself lighter and easier to lift over the top. Picking its own ball back up again on the right, it helps to pull the right side down.
The analysis based on friction is somewhat superficial, however. One could understand friction perfectly well and yet imagine the following situation. Astronauts bring back a piece of magnetic ore from the moon which does not behave like ordinary magnets. A normal bar magnet, c/1, attracts a piece of iron essentially directly toward it, and has no left- or right-handedness. The moon rock, however, exerts forces that form a whirlpool pattern around it, 2. NASA 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
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 normally obey but because, like the alchemists, they have discovered a very deep and fundamental principle of nature which forbids certain 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 laboratory 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 account. 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 quantity. 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 in or out through the doors.
Conservation of mass example 1 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.
c / A mysterious moon rock makes a perpetual motion machine.
In general, the amount of any particular substance is not conserved. 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 balls. 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
d / Example 1.
Section 1.2
Energy
15
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. The continual buildup in the “heat account” leaves less and less for the “motion account” and “height account,” causing the machine eventually to run down. These insights can be distilled into the following basic principle of physics: the law of conservation of energy It is possible to give a numerical rating, called energy, to the state of a physical system. The total energy is found by adding up contributions from characteristics of the system such as motion of objects in it, heating of the objects, and the relative positions of objects that interact via forces. The total energy of a closed system always remains constant. Energy cannot be created or destroyed, but only transferred from one system to another. The moon rock story violates conservation of energy because the rock-cylinder and the ball together constitute a closed system. Once the ball has made one revolution around the cylinder, its position relative to the cylinder is exactly the same as before, so the numerical energy rating associated with its position is the same as before. Since the total amount of energy must remain constant, it is impossible for the ball to have a greater speed after one revolution. If it had picked up speed, it would have more energy associated with motion, the same amount of energy associated with position, and a little more energy associated with heating through friction. There cannot be a net increase in energy.
Converting one form of energy to another example 2 Dropping a rock: The rock loses energy because of its changing position with respect to the earth. Nearly all that energy is transformed into energy of motion, except for a small amount lost to heat created by air friction. Sliding in to home base: The runner’s energy of motion is nearly all converted into heat via friction with the ground. Accelerating a car: The gasoline has energy stored in it, which is released as heat by burning it inside the engine. Perhaps 10% of this heat energy is converted into the car’s energy of motion. The rest remains in the form of heat, which is carried away by the exhaust. Cruising in a car: As you cruise at constant speed in your car, all the energy of the burning gas is being converted into heat. The tires and engine get hot, and heat is also dissipated into the air through the radiator and the exhaust.
16
Chapter 1
Conservation of Energy
Stepping on the brakes: All the energy of the car’s motion is converted into heat in the brake shoes. Stevin’s machine example 3 The Dutch mathematician and engineer Simon Stevin proposed the imaginary machine shown in figure e, which he had inscribed on his tombstone. This is an interesting example, because it shows a link between the force concept used earlier in this course, and the energy concept being developed now. The point of the imaginary machine is to show the mechanical advantage of an inclined plane. In this example, the triangle has the proportions 3-4-5, but the argument works for any right triangle. We imagine that the chain of balls slides without friction, so that no energy is ever converted into heat. If we were to slide the chain clockwise by one step, then each ball would take the place of the one in front of it, and the over all configuration would be exactly the same. Since energy is something that only depends on the state of the system, the energy would have to be the same. Similarly for a counterclockwise rotation, no energy of position would be released by gravity. This means that if we place the chain on the triangle, and release it at rest, it can’t start moving, because there would be no way for it to convert energy of position into energy of motion. Thus the chain must be perfectly balanced. Now by symmetry, the arc of the chain hanging underneath the triangle has equal tension at both ends, so removing this arc wouldn’t affect the balance of the rest of the chain. This means that a weight of three units hanging vertically balances a weight of five units hanging diagonally along the hypotenuse. The mechanical advantage of the inclined plane is therefore 5/3, which is exactly the same as the result, 1/ sin θ, that we got before by analyzing force vectors. What this shows is that Newton’s laws and conservation laws are not logically separate, but rather are very closely related descriptions of nature. In the cases where Newton’s laws are true, they give the same answers as the conservation laws. This is an example of a more general idea, called the correspondence principle, about how science progresses over time. When a newer, more general theory is proposed to replace an older theory, the new theory must agree with the old one in the realm of applicability of the old theory, since the old theory only became a accepted as a valid theory by being verified experimentally in a variety of experiments. In other words, the new theory must be backward-compatible with the old one. Even though conservation laws can prove things that Newton’s laws can’t (that perpetual motion is impossible, for example), they aren’t going to disprove Newton’s laws when applied to mechanical systems where we already knew Newton’s laws were valid.
e / Example 3.
Discussion Question
A Hydroelectric power (water flowing over a dam to spin turbines) appears to be completely free. Does this violate conservation of energy? If not, then what is the ultimate source of the electrical energy produced by a hydroelectric plant? B How does the proof in example 3 fail if the assumption of a frictionless surface doesn’t hold?
Discussion question A. The water behind the Hoover Dam has energy because of its position relative to the planet earth, which is attracting it with a gravitational force. Letting water down to the bottom of the dam converts that energy into energy of motion. When the water reaches the bottom of the dam, it hits turbine blades that drive generators, and its energy of motion is converted into electrical energy.
Section 1.2
Energy
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1.3 A Numerical Scale of Energy
Energy comes in a variety of forms, and physicists didn’t discover all of them right away. They had to start somewhere, so they picked one form of energy to use as a standard for creating a numerical energy scale. (In fact the history is complicated, and several different energy units were defined before it was realized that there was a single general energy concept that deserved a single consistent unit of measurement.) One practical approach is to define an energy unit based on heating water. The SI unit of energy is the joule, J, (rhymes with “cool”), named after the British physicist James Joule. One Joule is the amount of energy required in order to heat 0.24 g of water by 1 ◦ C. The number 0.24 is not worth memorizing. Note that heat, which is a form..."
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