Norris Notes

MEASURING and CALCULATING

Jack H. Norris
jacknorris1@mac.com
last update 4/13/08
I. There are three basic requirements for measurement.
  1. We should know what we are trying to measure.
  2. We should have a standard to compare with.
  3. We should have a method to make a comparison.
 II.  TERMS- (for measuring and calculating)
  1. Qualitatively--described with words
  2. Quantitatively--description based on measurement
III.  INTERNATIONAL SYSTEM (SI)
  1. Modern version of metric system.   KNOW the units and prefixes that are listed in Tables [2.1, 2.2, 2.3, and 2.4]. Be sure to be aware of the meanings of the prefixes in 2.4.
IV.  MASS
  1. WEIGHT --measure of force of gravity
  2. MASS --measurement that does not change from place to place
    1. Measure of quantity of matter in an object --- The international standard for mass is the prototype kilogram (kg)
    2. Instruments used to determine the mass of an object
      • gravitational balance
      • inertial balance
    3. The GRAM (1 g) is more commonly used to measure mass.
V.   LENGTH
  1. Length is the distance covered by a line segment connecting two points. Ex. A.____________B.
  2. The SI unit for length is the meter (M).
VI.  TIME
  1. Time is the interval between two occurrences.
  2. The SI unit for time is the second (s).
    1. There are 86,400 seconds in an average day.
    2. There are 3600 seconds in an hour.
    3. There are 60 seconds in a minute.
VII.  TEMPERATURE and HEAT
  1. Heat
    1. Heat is a form of energy.
    2. Heat flows from areas of high temperature to lower temperature (hot to cold).
  2. Temperature
    1. Temperature is a measure of the average kinetic energy per molecule. Some people say it is a measure of heat intensity.
      • Thermometer --instrument for measuring temperature
      • Types of thermometers used in class --alcohol AND computer probe   Mercury thermometers should be avoided. Mercury is toxic.
      • SCALES
        • FAHRENHEIT (0F) - freezing point for water is 32 degree F at 1 atm of pressure - boiling point for water is 212 degree C at 1 atm of pressure.
        • CELSIUS (0C) - freezing point for water is 0 degree C at 1 atm of pressure - boiling point for water is 100 degree C at 1 atm of pressure.
        • KELVIN (K) - freezing point for water is 273.15 Kelvin at 1 atm of pressure, boiling point for water is 373.15 Kelvin at 1 atm of pressure. There is no negative Kelvin.  One does not say"degree Kelvin." You should simply say Kelvin.
        • ABSOLUTE ZERO is the coldest temperature possible. It is zero K. At zero K all molecular motion (no average kinetic energy per molecule) has ceased.
VIII.  RELIABILITY
  1. Accuracy --
    1. Accuracy is how close a measurement is to the true or accepted (correct) value for the quantity. [The quality of a measuring instrument is probably the most important factor relative to accuracy.]
  2. Precision
    1. Precision is how close a set of measurements for a quantity are to one another. [Uncertainty in measurement]   The closer several measurements are to each other the more precise the measurements are considered to be.
X.   SIGNIFICANT DIGITS (Figures)
  1. Digits that occupy places for which actual measurement was made. The first guess is significant.
  2. Digits are used to indicate the exactness of a measurement.
  3. The last digit of a measurement is considered an estimate.
  4. RULES
    1. Digits other than zero are always significant.
    2. Zeros that are between nonzero digits are significant.
      • 7004  = 4 significant digits
      • 2.509 = 4 significant digits
    3. Zeros at the end of a number and to the right of a decimal point are always significant.
      • 4.7200 = 5 significant digits
      •  82.0    = 3 significant digits
    4. zeros between two other significant digits are always significant
      • 5.029         = 4sd
      • 306            = 3sd
      • 1.001         = 4sd
      • 234005.00 = 8sd    Make sure you understand why.
      • 102            = 3sd
    5. Zeros used solely for spacing the decimal point are not significant. The zeros are placeholders only.
      • 7000             = 1 significant digit
      • 0.00783        = 3 significant digit
      • 10.0023        = 6 significant digit
      • 5,360,000     = 3 significant digit
      • 6,360,000.0  = 8 significant digit
    6. Unlimited number of significant figures for --
      • counts   EX.    My mother had 19 siblings.
      • defined quantities     EX. 60 min = 1 h or 100 cm = 1 m8 significant digit
    7. Rules for rounding.
 
XI.  SCIENTIFIC NOTATION
  1. Convenient system of expressing very large or very small numbers
  2. Expresses all numbers as a product of a number between 1 and 10 and a whole number power of 10.
  3. In ADDITION AND SUBTRACTION --the answer may contain only as many decimal places as the measurement having the least number of decimal places.
    1. 677.1 + 39.24 + 6.232 would round to 722.6
    2. When adding numbers, the EXPONENTS MUST BE THE SAME.   Add & subtraact the coefficients, keep base & exponent; then rearrange in sci. notation if needed.   EXAMPLE = 5.6 x 103 - 6.1 x 103 = -0.5 x 103 = -5.0 x 102
    3. Additional discussion and examples.
  4. In MULTIPLICATION AND DIVISION, the answer may contain only as many significant digits as the measurement with the least number of significant digits.
    1. 1.13 X 5.126122 would round to 5.79 because the number 1.13 has three significant and 5.79 has three significant digits.
    2. When you multiply numbers in sci. notation, you MULTIPLY the COEFFICIENTS and ADD the exponents.
    3. When you divide numbers in sci. notation, you divide the coefficients and subtract the exponent in the denominator (bottom) from the exponent in the numerator.   EXAMPLE = 2.8 x 105 / 1.4 x 103 = (2.8/1.4) x 10(5-3) = 2.0 x 102
    4. Additional discussion and examples.
XII.  DERIVED UNITS
  1. Derived units are obtained by combining SI base units.
    1. Ex-- Distance divided by time equal SPEED, d/t=speed
    2. Ex-- AREA is length times length A=L x L
    3. Ex-- Volume is meter x meter x meter = m3
  2. Combined units combine base units. Be certain you know what the SI Base Units of Measurement are.   The SI symbols for the seven SI base units are: m, kg, s, A, K, mol, and cd.
    1. 1000 cm3 = 1000 ml = 1 L = 1 dm3
XIII.  CONVERSION FACTORS   -- Dimensional Analysis
  1. Ratios with a value equivalent to one
  2. Ex-- 1 dm3 /1000 cm3 = 1
  3. The value of any quantity multiplied by 1 is unchanged.
  4. The vertical bar eliminates the need for parentheses in conversion factors.
XIV.  FACTOR - LABEL
  1. Unit labels may be treated as factors and divided out.
  2. Most conversion factors have an unlimited number of significant digits because they are definitions.
    1. Ex 1 foot = 12 inches, you could put a thousand zeros after the 12 or no zeros.
  3. Unit cancellation marks act as a check as to whether a problem has been set up correctly. If you can use units to check your work you are not certain you are solving correctly.
XV.  DENSITY
  1. DENSITY is mass per unit of volume.  Density is the ratio of the mass of an object to its volume.
  2. The density of solids and liquids is generally measured in g/cm3
  3. D=m/V       Density= mass/volume
 
XVI.  SPECIFIC GRAVITY
  1. SPECIFIC GRAVITY (SG) is a comparison of the density of a substance to the density of a substance that is used as a reference. WATER AT 4 ºC is usually used as the reference.
  2. Since SG = density of substance in g/cm3 / density of water in g/cm3, the units cancel. Ergo, a measurement of density has NO UNITS.
  3. The SG of a liquid can be measured with a HYDROMETER.
 
XVI.  EVALUATING MEASUREMENTS
  1. ALL measurements contain error.   Even your grade computation.
  2. The ACCEPTED VALUE is the true or correct value based on references that are reliable.
  3. The EXPERIMENTAL VALUE is the measured value that is determined in the experiment.
  4. The ERROR is the difference between the accepted value and the experimental value. REMEMBER         ERROR = ACCEPTED VALUE - (minus) EXPERIMENTAL VALUE
  5. % ERROR= ( |error| / accepted value ) X 100%     In calculating % error the ABSOLUTE VALUE of the error is used.   Ergo, the % error will always be a positive value.
 
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