Topic Summaries: Foundations

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© 1998-2008 by Glenn Elert -- A Work in Progress
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  1. Units
    1. International System
      • The International System of Units is currently the generally accepted system of units in the physical sciences.
        • The international abbreviation of the name is SI (from the French Le Système International d'Unités).
      • The SI model has three major components.
        1. Seven well-defined, dimensionally independent, base units that are assumed irreducible by convention (meter, kilogram, second, ampère, kelvin, mole, and candela).
        2. A large number of derived units formed by combining base units according to the algebraic relations of the corresponding quantities (some of which are assigned special names and symbols and which themselves can be further combined to form even more derived units).
          • The derived units are coherent in the sense that they are all mutually related only by the rules of multiplication and division with no numerical factor other than 1 needed.
          • The derived units are also complete in the sense that one and only one unit exists for every defined physical quantity. Although it is possible to express many units in more than one way, they are all equivalent. (The converse statement is not necessarily true, however. Some units are used for more than one physical quantity.)
        3. Twenty currently agreed upon prefixes that can be attached to any of the base units or derived units with special names creating multiples and division as needed. (The exception to this rule is the kilogram, which is already itself a multiple of the gram. In this case, prefixes should be added to the word gram.)
          • The first three named mutiples are the first three powers of ten (101, 102, 103).
            Subsequent named multiples are larger than the previous named multiple by three orders of magnitude (106, 109, 1012, … ).
          • The first three named divisions are the first three negative powers of ten (10−1, 10−2, 10−3).
            Subsequent named divisions are smaller than the previous named division by three orders of magnitude (10−6, 10−9, 10−12, … ).
      • Other scientific, traditional, and practical units and unit systems are still in use and are still useful.
    2. Gaussian System
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    3. English System
      • traditional english units
        • length
        • mass
          • avoirdupoids
          • troy
        • area
        • volume
          • liquid
          • dry
          • cubic
      • non-metric scientific units
        • foot-pound-second system
          • pound force and slug
          • pound mass and poundal
        • and the rest
    4. Miscellaneous Units
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    5. Time
      • Time is a measure of the interval between two events.
      • Time is a cultural construct whereby an event can be associated with a series of numbers.
      • The SI unit of time is the second [s].
    6. Unit Conversion
      • Blah blah dimensional analysis blah blah.
        • When quantities are multiplied, divided, or raised to a power their units are likewise multiplied, divided, or raised to a power.
        • Quantities may be added or subtracted only if they have the same unit.
      • Blah blah factor label method blah blah.
        • multiply by a ratio equal to one with the intent of canceling the original unit(s)
  2. Measurement
    1. Significant Digits
      • Precision …
        • is the degree to which the results of several measurements agree with one another
        • is the exactness of a device
        • is determined by the place value of the last recordable digit (often referred to as the number of decimal places).
      • Accuracy …
        • is the degree to which the result of one measurement (or one computation based on several measurements) agrees with its true value
        • is the exactness of a measurement (or computation based on measured values)
        • is determined by counting the number of significant digits
      • The significant digits (or significant figures) in a measurement …
        • are those that are a part of the measurement
        • does not include placeholder zeroes
        • must be written even if the measurement …
          • is stated in standard form scientific notation
          • is converted to units that are decimal multiples or divisions of the original (for example, km, m, mm, etc.)
         
        Counting Significant Digits
        digit location significant?
        non-zero anywhere yes
        zero initial   no
        " medial   yes
        " final after the decimal point yes
        " " before a written decimal point yes
        " " before an unwritten decimal point maybe
         
      • Arithmetic using significant digits
        • Addition & Subtraction
          • The answer is only as precise as the least precise measurement.
        • Multiplication, Division, Powers & Roots
          • The answer is only as accurate as the least accurate measurement.
        • Numbers that are a part of a mathematical equation were never measured and therefore cannot affect the accuracy of a computation.
          • Rational numbers (1, 2, 3, ½, ⅔, etc.) are "perfect numbers" in theory and practice.
          • Irrational numbers (√2, π, e, etc.) …
            • cannot be written in decimal form using a finite number of digits
            • are only as accurate as the number of digits used for computational purposes
            • are effectively "perfect numbers" on a calculator since the number of digits returned is greater than that of nearly every measurement ever made.
              • The π button on calculator gives so many digits that it does not affect the accuracy of most computations.
        • The formally stated results of any computation based on measured values should be stated with an appropriate number of significant digits.
          • Once the necessary number of significant digits is determined, identify the last digit to be recorded.
            • Add one to this digit if the next digit is 5 or greater; that is, round up.
            • Do nothing to this digit if the next digit is 4 or less; that is, round down.
          • Any necessary rounding should be done only after computation is completely finished.
            • Never round the results of a partially calculated value.
            • If the results of one computation are to be used in another, use the unrounded value of the first computation (if it is available).
    2. Orders of Magnitude
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  3. Graphs
    1. Graphical Representation of Data
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    2. Linear Regression
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    3. Curve Fitting
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    4. Calculus
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  4. Vectors
    1. Scalars, Vectors, Tensors
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    2. Vector Addition & Subtraction
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    3. Vector Resolution & Components
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    4. Vector Multiplication
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  5. Miscellaneous Pages
    1. Symbols Used in This Book
    2. Physical Constants
    3. Astronomical Data
    4. Periodic Table of the Elements
    5. Personalities in Phyiscs
    6. Nobel Prize Winners in Physics

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