v3.1.6 / chapter 14 of 14 / 01 jan 08 / greg goebel / public domain
* 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)
Mechanical advantage:
force1 * displacement1 = force2 * displacement2
Gravitational force:
gravitational_force = 6.672E-11 * mass1 * mass2 / distance^2
Electrical force:
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^2
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* Angular velocity:
angular_velocity = linear_velocity / ( 2PI * radius )
Torque:
torque = force * radius
Moment of inertia:
moment_of_inertia = SUM( mass * radius^2 )
Angular momentum:
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:
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 )^1.5
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* 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 )
Wave parameters:
frequency = velocity / wavelength
period = 1 / frequency = wavelength / velocity
Sinusoidal wave function:
amplitude * SIN( 360 * time / period + phase )
Doppler shift:
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 )
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* Length:
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:
1 square centimeter = 0.16 square inches
1 square meter = 1.2 square yards = 10.8 square feet
1 hectare = 2.471 acres = 10,000 square meters
1 square kilometer = 0.386 square miles = 100 hectares
1 square inch = 6.45 square centimeters
1 square foot = 929 square centimeters
1 square yard = 0.836 square meters
1 acre = 4,047 square meters = 0.405 hectares
1 square mile = 2.59 square kilometers = 259 hectares = 640 acres
* Volume:
1 cubic centimeter = 0.061 cubic inch
1 liter = 1.06 quarts = 0.26 gallon = 0.1 cubic meter
1 cubic meter = 1.308 cubic yards = 35.31 cubic feet
1 cubic inch = 16.4 cubic centimeters
1 cubic foot = 0.028 cubic meters
1 cubic yard = 0.765 cubic meters
1 pint = 0.47 liters
1 quart = 1.06 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 cubic mile = 4.17 cubic kilometers
* Mass:
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
* Temperature:
degrees Celsius = (5/9) * ( F - 32 )
degrees Fahrenheit = (9/5) * C + 32
degrees Kelvin = C + 273.15
0 K = -459.67 F
* Energy:
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
* Power:
1 horsepower = 745.7 watts
1 kilowatt = 1.341 horsepower
* Pressure:
1 atmosphere = 101,325 pascals = 760 torr (mm hg) = 14.7 PSI
1 bar = 100,000 pascals
* Miscellaneous:
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.
* 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, given a proper understanding of them, predictable. It must be added that the notion that our senses perceive a logically consistent Universe around us is an assumption -- a concept known as "scientific objectivism" or just "objectivism". As was pointed out by Bishop Berkeley and some of his contemporaries, how can we prove that our senses aren't 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.
Berkeley's "solipsism" comes up in fantasy stories every now and then, for example the movie THE MATRIX was more or less based on the idea: "Is it live or is it Memorex?" The answer to Berkeley's ingenious proposition is that there is no way to prove that the Universe "out there" isn't an illusion -- but so what? Whatever the actual nature of the reality around us, there is no evidence so far to contradict the individual or collective belief that it operates by consistent rules that make no conscious concessions to our convenience. If we are really stuck in an elaborate virtual-reality game, we're still stuck with trying to figure out the way that game operates; and even if the Universe is an illusion, it still can inflict pain on us if we don't pay attention to the rules.
In the end, Berkeley's solipsism runs into the roadblock of Occam's razor: we can choose between the assumption that there is a Universe "out there" as determined by the evidence of our senses -- or the awkward possibility that there is an indefinite set of possibilities that "something else" is going on, with no evidence at all to permit us to figure out what these alternatives are, much less which of them to select. This isn't much of a choice. In other words, if scientific objectivism is an assumption, it's a far less burdensome and much more workable assumption than the alternative.
The most that can be said of Berkeley's solipsism is that it is useful to keep an eye out for possible contradictions in our perceptions of the operation of the Universe "out there" that might indicate a need to reexamine our assumptions. This is useful even if the notion of scientific objectivism is absolutely taken for granted, since it's a basic principle of the sciences that our senses can be misleading -- which is why the sciences demand duplicable observations or experiments.
* 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 criticised her belief that it was based on strictly objective principles. Perfectly coherent arguments could be constructed for or against many of her assertions as suited the subjective viewpoints of those willing to debate the matter, with the sciences providing little or no capability to clearly choose between them. 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, mysticism, and religion into the trashbin.
While all legitimate scientists accept scientific materialism, there is a loud clash between advocates of philosophical materialism, particularly those engaged in attacks on religion, and those who suggest that the realm of the objective and proveable is simply too narrow and limited to be adequate to provide a basis for the sensible conduct of our lives. The dubious nature of Rand's attempt to construct a social philosophy on an "objective" basis does suggest that there are limits to objectivism / materialism.
The hairsplitting subject of objectivism / materialism amounts to something of an obnoxious and annoying 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 short-tempered Johnson:
BEGIN QUOTE:
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]
* The evolution of this document has been complicated, to put it mildly. 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 in a simple fashion. It was a surprisingly difficult document to write, since I had to balance comprehensiveness against length, detail against the need to provide clear explanations of things, 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 references to 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 very 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!"
* Sources include:
I browsed through the Microsoft ENCARTA interactive encyclopedia to provide many of the additional details included in the v2.2.0 version. As it turned out, ENCARTA is not only readable but covers topics in physics, such as turbulent and incompressible flows, that aren't even mentioned in some of the first-year physics texts I've poked through. 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 / gvg / Single-file document on elementary mechanics.
v2.0.0 / 01 apr 02 / gvg / 8 chapter survey document.
v2.1.0 / 01 apr 04 / gvg / Various enhancements, built up to 11 chapters.
v2.1.1 / 01 may 04 / gvg / Minor tweaks and corrections.
v2.2.0 / 01 sep 04 / gvg / Extensive "modern physics" oriented changes.
v2.2.1 / 01 feb 05 / gvg / Some follow-up tweaks & corrections.
v3.0.0 / 01 jul 05 / gvg / General enhancement, up to 12 chapters.
v3.1.0 / 01 jan 06 / gvg / Tweaks, more on materials, up to 14 chapters.
v3.1.1 / 01 jun 06 / gvg / More tweaks!
v3.1.2 / 01 nov 06 / gvg / Still more tweaks!
v3.1.3 / 01 jan 07 / gvg / A few minor bug fixes.
v3.1.4 / 01 apr 07 / gvg / Fixes on lift and evolutionary thermodynamics.
v3.1.5 / 01 aug 07 / gvg / Cleanups here and there.
v3.1.6 / 01 jan 08 / gvg / Comments on objectivism, fixes on bridges.
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