* This chapter gives summaries of useful formulas, and a set of useful conversion factors and other data, with most values given to three digits of precision.
* Velocity of a mass falling near the surface of the Earth, where the acceleration of gravity is 9.81 meters per second squared:
velocity = acceleration * time = 9.81 * time
Displacement of a mass falling near the surface of the Earth:
displacement = (1/2) * 9.81 * time^2
Definitions of force, momentum, energy, and power:
momentum = mass * velocity (Newton's First Law of Motion) force = mass * acceleration (Newton's Second Law of Motion) = delta_momentum / delta_time (alternate form) energy = power * time power = energy / time
Potential energy of a mass raised to a height near the surface of the Earth:
potential_energy = 9.81 * mass * height
Kinetic energy of a mass:
kinetic_energy = (1/2) * mass * velocity^2 kinetic_energy = ( momentum^2 ) / ( 2 * mass )
Conservation of momentum:
mass1 * velocity1 = mass2 * velocity2 (Third Law of Motion)
force1 * displacement1 = force2 * displacement2
gravitational_force = 6.672E-11 * mass1 * mass2 / distance^2
electrical_force = 8.9875E9 * charge1 * charge2 / distance^2
(Incompressible) fluid flow:
fluid_flow_rate = cross_section1 * velocity1 = cross_section2 * velocity2
* Structural formulas:
elastic_modulus = stress / strain -- OR: strain = stress / elastic_modulus force = stress * strain (Hooke's law) strain_energy = volume * ( 1/2 ) * stress * strain^2BACK_TO_TOP
* Angular velocity:
angular_velocity = linear_velocity / ( 2PI * radius )
torque = force * radius
Moment of inertia:
moment_of_inertia = SUM( mass * radius^2 )
angular_momentum = moment_of_inertia * angular_velocity
Work and kinetic energy in rotating systems:
work = torque * angular_displacement kinetic_energy = (1/2) * moment_of_inertia * angular_velocity^2
centripetal_acceleration = radius * angular_velocity^2
Kepler's Third Law:
orbital_radius^3 = constant * orbital_period^2
The constant depends on the body being orbited and has a value of roughly 131,000 for the Earth, and giving the related formulas:
orbital_radius = 5,078 * orbital_period^(2/3) orbital_period = ( orbital_radius / 5,078 )^1.5BACK_TO_TOP
* Ideal gas law:
pressure1 * volume1 pressure2 * volume2 --------------------- = --------------------- temperature1 temperature2
(Thermal) definition of entropy:
heat_transfer entropy = ---------------------- absolute_temperature
Efficiency of an engine:
heat_out efficiency = 1 - ---------- heat_in
Efficiency of a Carnot cycle engine:
efficiency = 1 - ( output_temperature / input_temperature )
frequency = velocity / wavelength period = 1 / frequency = wavelength / velocity
Sinusoidal wave function:
amplitude * SIN( 360 * time / period + phase )
Tm = T * ( 1 +/- fraction_soundspeed )
Period of a pendulum:
period = 2PI * SQRT ( acceleration_of_gravity / length_of_rod )
Speed of light: 300,000,000 meters per second = 186,000 miles per second
Index of refraction & Snell's law:
speed_of_light_in_vacuum index_of_refraction = ---------------------------- speed_of_light_in_material R1 / R2 = SIN( angle1 ) / SIN( angle2 )BACK_TO_TOP
1 Angstrom = 10^-10 meter 1 micron = 10^-6 meter 1 centimeter = 0.394 inch 1 meter = 39.37 inches = 3.28 feet = 1.09 yards 1 kilometer = 0.632 mile = 3,280 feet = 1094 yards 1 inch = 2.54 centimeters (exactly) 1 foot = 30.5 centimeters = 0.305 meters 1 yard = 0.914 meters = 91.4 centimeters 1 mile = 1.61 kilometers 1 nautical mile = 6,076 feet = 1.15 mile = 1.85 kilometers 1 light-year = 9.46E12 kilometers = 5.88E12 miles 1 parsec = 3.26 light-years
* Angular velocity:
RPM = 60 * radians / 2PI degrees per second = 360 * radians / 2PI
* Area ("sq" means "square"):
1 sq_centimeter = 0.16 sq_inches 1 sq_meter = 1.2 sq_yards = 10.8 sq_feet 1 hectare = 2.471 acres = 10,000 sq_meters 1 sq_kilometer = 0.386 sq_miles = 100 hectares = 247.1 acres 1 sq_inch = 6.45 sq_centimeters 1 sq_foot = 929 sq_centimeters 1 sq_yard = 0.836 sq_meters 1 acre = 4,047 sq_meters = 0.405 hectares 1 sq_mile = 2.59 sq_kilometers = 259 hectares = 640 acres
* Volume ("cu" means "cubic"):
1 cu_centimeter = 0.061 cu_inch 1 liter = 61.02 cu_inches = 0.26 gallon = 0.001 cu_meter 1 cu_meter = 1.308 cu_yards = 35.31 cu_feet = 1,000 liters 1 cu_inch = 16.4 cu_centimeters 1 cu_foot = 0.0283 cu_meters = 28.3 liters 1 cu_yard = 0.765 cu_meters 1 pint = 0.474 liters 1 quart = 0.946 liters 1 US gallon = 3.79 liters 1 imperial gallon = 1.21 US gallons = 4.546 liters 1 barrel (oil) = 42 US gallons = 158.98 liters 1 cu_mile = 4.17 cu_kilometers
1 kilogram = 1,000 grams = 2.205 pounds 1 tonne = 1,000 kilograms = 2,204.6 pounds = 1.10 tons 1 ounce = 28.34 grams 1 pound = 16 ounces = 0.454 kilogram 1 ton = 2,000 pounds = 907 kilograms
degrees Celsius = (5/9) * ( F - 32 ) degrees Fahrenheit = (9/5) * C + 32 degrees Kelvin = C + 273.15 0 K = -459.67 F
1 electron-volt = 1.602 * 10^-26 joule 1 erg = 10^-7 joule 1 foot-pound = 1.356 joules 1 calorie = 4.184 joules 1 btu = 1,055.6 joules = 252.2 calories 1 kilowatt-hour = 3.6 * 10^6 joules
1 horsepower = 745.7 watts 1 kilowatt = 1.341 horsepower
1 atmosphere = 101,325 pascals = 760 torr (mm hg) = 14.7 PSI 1 bar = 100,000 pascals
km per liter = 0.425 miles per gallon seconds in day = 86,400 seconds in year = 31,536,000
* A few inexact rule-of-thumb conversions, useful for quick and rough calculations:
2 inches = 5 centimeters 1 foot = 30 centimeters 1.1 yards = 1 meter 5 miles = 8 kilometers 500 pounds = 225 kilograms 1,000 pounds = 450 kilograms 1 ton = 900 kilograms 1.1 tons = 1 tonne 1 quart = 1 liter degrees Celsius = 2 * (degrees Fahrenheit - 30) degrees Fahrenheit = 30 + degrees Celsius / 2
The temperature conversions are particularly inexact, but useful for figuring out what the weather's like outside.BACK_TO_TOP
* The discussion of the nature of physics in the first chapter described science in general and physics as an effort to understand the rules of the operation of the Universe, with these rules being fixed and, when they're understood, predictable. This is the basic premise of the sciences, but it should be pointed out that it is based on an ultimately unprovable assumption.
That assumption was most astutely observed by the Scots scholar David Hume (1711:1776), who posed the question of what would happen if one billiard ball struck another. Of course, that's a simple question in physics, and it's very easy to predict what should happen with fair confidence. Hume knew that, but said that the only way we know so is by considerable observation of the collisions of billiard balls, or the operation of similar systems. If we didn't have such experience, would we have any way of knowing what would happen?
Hume's answer was simple: NO. We don't have the rule book to nature; all we know about it is what we have observed; our observations of past events are incomplete, and we have no way of knowing if events in the future will be the same as they were in the past. Nature behaves as it does and any expectations we might have on how it should behave -- when we don't have any experience to know -- are irrelevant. All we can do is observe how billiard balls behave in collisions and construct an abstract model on how they should behave in the future.
That model is an inference; Hume noted that all we can observe is correlations, such as the effect of one billiard ball on another, we can't actually observe cause, we just infer it from correlations. We then use the model to predict future actions of billiard balls. The model has no more validity than the conformance of its predicted results to observations.
Anything in the model that is not linked to observations is a necessary accounting device at best; excess baggage at worst. Yes, we are inclined to assume, as per Dick Feynmann, that nature does operate by well-defined rules that we can uncover through research, that we can indeed discover "the rules of the game", but our models will always be subordinate to observations. If our observations aren't supported by the model, we have to change the model.
This is reasonable, but it is based on the unproveable assumption that the Universe operates by orderly and predictable rules, a notion referred to as "scientific objectivism". Going even beyond that, Hume also pointed to the observation of his contemporary, Bishop Berkeley, that we can't even prove that our senses are not misleading us. Maybe there isn't a Universe "out there", it's just an illusion. Maybe there isn't even more than one individual in the Universe, and the perception of other individuals is part of that individual's illusion -- a notion referred to as "solipsism".
Having reached this limit, however, Hume, an entirely practical person, dismissed it as ridiculous. Nobody sane sincerely questions the existence of an external reality; we simply can't do so, it is fundamental for getting through our lives. We also assume the Universe is regular in its operations, since otherwise we wouldn't be able to learn anything from experience. We cannot prove there is an objective reality, and that the real Universe operates by regular rules -- but we have no reason to question those assumptions, and have nowhere to go if we do so.
We cannot prove that the Sun will rise tomorrow, all we can say that it always has, but nobody has any real doubt that it will. Nobody with any sense would try to argue it won't, and anyone who did would not be able to show what the point of doing so was.
Hume was the first to admit that even if our observations of the Universe don't come with a guarantee, they're the only way we have of knowing how the Universe actually works. He recognized that in practice the sciences work very well, and would have no sympathy with modern cranks who try to use his work to attack the sciences:
If we take in our hand any [work claiming to establish the factual] ... let us ask: Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames, for it can contain nothing but sophistry and illusion.
Hume's only real point in his considerations of the limits of the sciences was to counsel caution: that we don't know any more about the Universe than our observations tell us; that we should be careful to make sure our observations are accurate; that we should be wary when we are drawing inferences from skimpy data; that that we should be aware that new observations could always come up to alter our understanding.
* The Russian-American author Ayn Rand (1905:1982) came up with her own variation on objectivism, which is worth mentioning here since the term is most strongly associated by the public with her writings. Her usage is somewhat confusing, because her focus wasn't on the sciences, to which she was largely indifferent; she believed that morals and society should be based on objective criteria, and using that viewpoint constructed a libertarian social philosophy.
However, even some of those sympathetic to her libertarian philosophy have criticized her belief that it was based on strictly objective principles. Formal logical reasoning is powerful within its own domain, but it is weak when confronted with serious problems in human societies that can't be reduced to a calculus. They cannot because, as Hume pointed out, moral questions -- what we see as right or wrong, in the broadest sense what we like or don't like -- are rooted in human custom and emotion. The facts don't tell us what we want, they simply tell us what our options are, and how to weigh them.
To be sure, we can apply sensibility to screen out obviously implausible solutions to such real-world problems, but which one of the plausible solutions is the best is, to a degree, a matter of opinion. We certainly cannot prove after the fact that a different solution might not have worked better. In addition, as Rand grew older, she constructed an egocentric and imperious "cult of personality" around herself that didn't suggest much in the way of objectivity.
* The term "scientific materialism" is also used in place of "scientific objectivism", mostly to make a distinction from the notion of "philosophical materialism". Philosophical materialism takes the idea of understanding the operation of the Universe in an objective fashion and extends it to asserting that no other approach is valid in general -- in effect tossing emotion, intuition, and mysticism into the trashbin. While all legitimate scientists accept scientific materialism, there is a loud clash between advocates of philosophical materialism and those who suggest that the realm of the objective and provable is simply too narrow and limited to be adequate to provide a basis for the sensible conduct of our lives.
That is, of course, an argument completely irrelevant to an introductory document on physics. The sciences are by definition about the factual, and whatever arguments can be made for the non-factual are of little direct utility to the sciences. Those who wish to argue the case for the significance of non-factual can do so elsewhere.
The faintly disreputable subject of objectivism / materialism amounts to something of a hairsplitting and obnoxious distraction to anyone who just wants to learn about physics, which is why it's discussed in a footnote here. The 18th-century man of letters Samuel Johnson (1709:1784) neatly expressed that irritation, as recorded by his biographer James Boswell (1740:1795) after he mentioned Berkeley's solipsism to the bearlike Johnson:
I observed, that though we are satisfied his doctrine is not true, it is impossible to refute it. I never shall forget the alacrity with which Johnson answered, striking his foot with mighty force against a large stone, till he rebounded from it -- "I refute it THUS!"
END QUOTE [emphasis added]BACK_TO_TOP
* This document has a complicated history. I wrote a document on interstellar flight that stated that Einstein's theory of relativity ruled out faster-than-light flight. Having said that, I realized I needed to write a document on relativistic physics to back that statement up.
Relativistic physics builds on classical physics, and to write about relativistic physics I needed to establish definitions from classical physics. However, as I wrote an introductory section to establish those definitions, I concluded that if I wrote other articles in physics I would need to establish the same basic definitions in them as well. I decided to write an independent document covering them to eliminate duplication.
The result was the v1.0 version of this document, which was short and simply discussed elementary concepts in classical mechanics. That was a useful start, but proved far from satisfactory, since it offered too little to attract readers and seemed far too limited for my own taste. The v2.0.0 version was a major expansion to multiple chapters that covered the fundamentals of classical physics. It was a surprisingly difficult document to write, since I had to balance comprehensiveness against length, detail against clarity, and still keep it coherent and interesting.
It has since been enhanced further; I keep finding new items to add to the toolkit. The v3.0.0 version actually started out as nothing more than an update for the image file format, but as I reviewed the document I realized that it needed much more work and I ended up heavily rewriting it.
I had quite a bit of fun, along with a comparable amount of frustration, writing this document, since much of it was written off the top of my head with occasional validations from textbooks. It's amazing how much information a person picks up in a lifetime and learns to take for granted. Writing it down shows how much there really is there, and also reveals embarrassing gaps in understanding. Explaining really simple ideas turns out to be difficult. It is also interesting that, even though this is a very basic physics document -- tuned for advanced high-schoolers, junior college students, or in general "people who don't do physics for a living" -- I keep discovering new insights into simple physics that drive continuous improvements in the text.
Trying to make it fun was another challenge, and I've ended up tapping a lifelong interest in cartoons, B-movies, science fiction, and other trivia to brighten up the text a bit. Some may object, but I'd like to make this document as entertaining to read as possible -- and besides sometimes the gag scenarios, such as the cartoon laws of physics, provide interesting counterexamples to highlight how things actually work in real world. If anybody wants a more formal text on physics, there's plenty of them in any reasonable public library. I will suggest, however, that this document does provide a useful complement to a formal physics text, since it gives plenty of useful examples to illustrate the basic principles of physics. Formal math is easier to handle if there's some understanding of what it actually means in practice.
* Having mentioned the cartoon laws of physics, I have to end with a scene from one of the more extreme Coyote versus Roadrunner cartoons. Wile E. Coyote finally manages to corner the Roadrunner on a narrow ledge hanging far out from a cliff, but as the Coyote stands there gloating, the part of the ledge under him gives way, to leave the tip of the ledge and the Roadrunner suspended in the air. After an amazingly complicated and sadistically brutal fall, the Coyote finally comes to rest. He gathers his wits and looks up from the desert floor to see the Roadrunner still standing on the broken ledge, suspended in the air. The Coyote holds up a sign that reads:
I WOULDN'T MIND -- BUT THAT DEFIES THE LAW OF GRAVITY.
The Roadrunner holds up a sign in reply:
THAT'S OK -- I NEVER STUDIED LAW.
-- and streaks off: "BeepBeep!"
* As concerns copyrights and permissions for this document, all illustrations and images credited to me are public domain. I reserve all rights to my writings. However, if anyone does want to make use of my writings, just contact me, and we can chat about it. I'm lenient in giving permissions, usually on the basis of being properly credited.
* Sources include:
I browsed through the Microsoft ENCARTA interactive encyclopedia to provide many of the additional details included in the v2.2.0 version. I used Web searches as well; the Wikipedia online encyclopedia was very useful, but in general I just picked up bits and pieces from here and there and fitted them together.
I also have to give some credit to the writings of David Macaulay. I didn't use them as sources here at all, but they were a stylistic influence, since this gadgety document could be considered as a halfway house between a typical elementary physics text and Macaulay's THE WAY THINGS WORK.
* Revision history:
v1.0 / 01 jul 99 / Single-file document on elementary mechanics. v2.0.0 / 01 apr 02 / 8 chapter survey document. v2.1.0 / 01 apr 04 / Various enhancements, built up to 11 chapters. v2.1.1 / 01 may 04 / Minor tweaks and corrections. v2.2.0 / 01 sep 04 / Extensive "modern physics" oriented changes. v2.2.1 / 01 feb 05 / Some follow-up tweaks & corrections. v3.0.0 / 01 jul 05 / General enhancement, up to 12 chapters. v3.1.0 / 01 jan 06 / Tweaks, more on materials, up to 14 chapters. v3.1.1 / 01 jun 06 / More tweaks! v3.1.2 / 01 nov 06 / Still more tweaks! v3.1.3 / 01 jan 07 / A few minor bug fixes. v3.1.4 / 01 apr 07 / Fixes on lift and evolutionary thermodynamics. v3.1.5 / 01 aug 07 / Cleanups here and there. v3.1.6 / 01 jan 08 / Comments on objectivism, fixes on bridges. v3.1.7 / 01 dec 08 / Fixes on thermodynamic examples. v3.1.8 / 01 oct 09 / A few minor fixes. v3.2.0 / 01 nov 10 / General cleanup, thermodynamics update. v3.2.1 / 01 oct 12 / Review & polish. v3.2.2 / 01 sep 14 / Review & polish. v3.2.3 / 01 aug 16 / Review & polish.BACK_TO_TOP