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Chapter

4 Chapter 4.  Unit Systems

 4.1 US System 4.2 Metric System 4.3 Unit Prefixes 4.4 Using Natural Phenomena 4.5 SI System (System International) 4.6 UM System (Unified Math) 4.7 Measuring Instruments 4.8 Calibration 4.9 Impact of Units of Formulas 4.10 Activities and Explorations

4.1

#### US System Abbrev. Name Derivation Illustrative Examples Length Dimension in inch between two knuckles of an average finger ft foot 12 in average man's foot yd yard 3 ft from floor to standard door knob mi mile 5280 ft four times around a traditional track field Weight Dimension oz ounce one-fourth cube of butter lb pounds 16 oz four cubes of butter ton tons 2000 ton a lot of butter Area Dimension in2 square inch square that surrounds a quarter ft2 square feet 144 in2 average size of floor tile yd2 square yard 9 ft2 an average card table acre acre 4840 yd2 just over 90 yards of a football field section section 640 acres 1 square mile township township 36 sections 6 miles by 6 miles Volume Dimension tsp teaspoon standard spoon size tbsp tablespoon 3 tsp large spoon size floz fluid ounce 6 tsp 2 large spoon sizes cup cup 8 floz small size carton of milk pt pint 2 cup standard size of carton for cream qt quart 2 pt medium size of carton for milk gal gallon 4 qt large size of carton for milk ` ` ` `

Common US Units

The US system and the British system from which it evolved seem to place a heavy emphasis on the ability to divide numbers evenly by small numbers.  For example, we can divide the number 12 evenly by 2, 3, and 4; the number 16 in half 4 times; and the number 5280 in half six times as well as by 3, 4, 5, and 6.

4.2

#### Metric System Abbrev. Name Derivation Illustrative Examples Length Dimension mm meter width of one dime cm centimeter 10 mm width of ten dimes m meter 10 cm one half height of 6 foot 6 in basketball player km kilometer 10 m almost three times around high school track field Weight Dimension mg milligram g gram 1000 mg water in small thumbnail (1 ml) kg kilogram 1000 g water in a common size of bottled water (1 liter) tonne metric ton 1000 kg water in a cubic meter container Area Dimension mm2 square millimeter size of surface of a pin head cm2 square centimeter 100 mm2 size of a small fingernail dm2 square deciimeter 100 cm2 size of a small fingernail m2 square meter 100 dm2 just larger than an average card table km2 square kilometer 1,000,000 m2 about one third of a square mile barn barn 10-28 m2 cross section of nuclei of atom Volume Dimension mL milliliter 1 cm3 small thumbnail of water L liter 1000 mL common size of bottled water ` ` ` `

Common Metric Units

The metric system emphasizes the use of the decimal system of numbers.  Notice that the derivations of related units come from factors of 10.  This makes unit conversions and calculations more convenient, it has become an international standard, especially for scientific publications.  Common conversions from other systems (including the US System) to the metric system can be found at websites such as:

http://www.chezcrowe.com/conv_app.htm (Conversion)

4.3

#### Unit Prefixes Name Prefix Unit Identity Inverse Unit Identity yotta Y 10^24 m/Ym 10^-24 Ym/m zetta Z 10^21 m/Zm 10^-21 Zm/m exa E 10^18 m/Em 10^-18 Em/m peta P 10^15 m/Pm 10^-15 Pm/m tera T 10^12 m/Tm 10^-12 Tm/m giga G 10^9 m/Gm 10^-9 Gm/m mega M 10^6 m/Mm 10^-6 Mm/m kilo k 10^3 m/km 10^-3 km/m hecto h 100 m/hm .01 hm/m deca da 10 m/dam .1 dam/m 1 deci d 0.1 m/dm 10 dm/m centi c 0.01 m/cm 100 cm/m milli m 10^-3 m/mm 10^3 mm/m micro µ 10^-6 m/µm 10^6 µm/m nano n 10^-9 m/nm 10^9 nm/m pico p 10^-12 m/pm 10^12 pm/m femto f 10^-15 m/fm 10^15 fm/m atto a 10^-18 m/am 10^18 am/m zepto z 10^-21 m/zm 10^21 zm/m yocto y `10^-24 m/ym` `10^24 ym/m`

Unit Prefixes

Since the metric system is conveniently based on the decimal system, we can use prefixes to create units related to a given unit by factors of ten.  For example, a kilogram (using the prefix "kilo") is a new unit derived from the unit gram by multiplying by 1000.  The above table gives the unit identities that convert between such units.

4.4

#### Using Natural Phenomena [meter~Length(Thing)] Calculated from curvature of the earth  [liter~Volume(Thing)] Cubic decimeter   [kilogram~Mass(Thing)] Liter of water   [gram~Mass(Thing)] Cubic centimeter of water

The metric system begins with a definition of  the unit "meter" for the dimension "length" based on the natural phenomena of the curvature of the earth which a person can determine using trigonometry to a reasonable amount of accuracy anywhere on the earth.  Later scientists gained more accuracy by using natural concepts of light.  The choice of meter came close to the commonly used unit of a yard.

The metric system defines the unit "liter" for the dimension "volume" as a cubic decimeter.  This particular choice of how a liter depends on a meter results in a volume unit close to the unit "quart".  Now using the natural phenomena of water, the metric system defines the unit "kilogram" of the dimension "mass" as the mass of water contained in one liter.  This means that a gram of water fills a cube having one-centimeter edges.

 Water is an abundantly available phenomena.  In addition to the mass unit Liter, we use water to define units of temperature (degrees Centigrade) and energy (calories).  Following the decimal goals of the metric system, the freezing point of water determines zero degrees Centigrade and the boiling point of water determines 100 degrees Centigrade.  The unit kelvin (K) uses the same increment (1°C = 1 K) but  shifts the value 0 to absolute zero.  Originally, scientists measured energy with the unit calorie defined as the amount of energy to raise one gram of water one degree Centigrade. Today, they use the unit joules to measure energy and the relationship between calories and joules for a particular substance (in addition to just water) is the specific heat of that substance. For  more information see http://library.thinkquest.org/C004970/thermo/specific.htm?tqskip1=1

4.5

#### SI System (System International) Abbreviation Unit Dimension SEVEN BASE UNITS m meter length kg kilogram mass s second time A ampere electric current K kelvin temperature mol mole amount of matter cd candela luminous intensity ANGLE UNITS rad radians plane angle sr steradians solid angle DERIVED UNITS m2 meters squared area m3 meters cubed volume m/s meters per second velocity m/s2 meters per second squared acceleration rad/s radians per second angular velocity rad/s2 radians per second squared angular acceleration kg/m3 kilograms per meters cubed density cd/m2 candela per meter squared luminance A/m ampere per meter magnetic field strength DERIVED UNITS WITH NEW SYMBOLS N = kg*m/s2 newton force Pa = N/m2 pascal pressure J = N*m joule quantity of energy W = J/s watt power V = W/A volt voltage = V/A ohm electric resistance C = A*s coulomb electric charge F = C/V farad electric capacitance Hz = cycles/s hertz frequency Wb = V*s weber magnetic flux T = Wb/m2 tesla magnetic flux density H = Wb/A henry inductance lm = cd*sr lumen flux of light lx = lm/m2 lux illumination

SI System

Common SI Units

The seven base units create the foundation from which we derive all other SI units.  Notice that each unit combines previously defined units using simple division and multiplications.  In fact, most of units relate back to the first three base units (meters, kilograms, and seconds) through simple algebraic substitutions.  A 75 watt light bulb refers to a 75 kg*m2/s3 light bulb where we have the chain of derivations: W=J/s=N*m/s= (kg*m/s2)*m/s= kg*m2/s3.  Even the base unit "ampere" relates back to the first three units through the relationship between magnetic forces and electronic currents; briefly, an ampere represents the amount of current in two straight parallel wires set 1 meter apart to produce a magnetic force of 2*10^-7 newtons. (See http://physics.nist.gov/cuu/Units/units.html)

The angle units represent supplementary units that use length and area to measure plane and solid angles respectfully.  You can find more information concerning the SI System of Units at sites such as:

http://metre.info/ (Extensive Details of SI or Modern Metric System)

http://www.metricmethods.com/metricmoments.html

4.6

#### UM System (Unified Math) Abbreviation Unit Dimension NINE BASE UNITS m meter length kg kilogram mass s second time A ampere electric current K kelvin temperature mol mole amount of matter cd candela luminous intensity dol dollar monetary value ins instance occurance EXAMPLE DERIVED UNITS dol/m dollars per lineal meter price per length ins/s instance per second event frequency s/ins seconds per instance duration dol/ins dollars per instance price dol/unit dollars per unit unit price

UM System

The Unified Math® system of units adds two new base units to the SI system, allowing us to handle business and statistical problems.

4.7

#### Measuring Instruments   Most measuring instruments relate values of a given dimension to the values in other dimensions.  Usually the other dimension is "length" with an associated a scale.  A traditional thermometer relates the expansion of some material (like mercury) in a tube that has been calibrated with a scale that measures the length that the material expands.  Consider modern cars, we relate the dimension "rotation" of the tires to the gauge in a speedometer where a needled moves a particular length and turns a counter for mileage.  Earlier, on some covered wagons, they had a device that counted the dimension "rotation of the wheel".  From that information, they could then calculate the approximate distance traveled.

4.8

#### Calibration    The process of calibrating a measuring instrument often involves adjusting the size of one dimension to match a predefined value of another dimension.  Take for example a kitchen scale that measures weight based on a spring inside the measuring instrument.  Such a device quite often has a knob (shown on the back of the one above) that allows the user to adjust the pointer connected to the spring so that it lines up with the 0 mark on the scale.

4.9

#### Impact of Units on Formulas Distance = Velocity * Time d = v * t   (if d miles, v miles/hour, t hour) d = v * t / 60  (if d miles, v miles/hour, t minutes) d = 1.61 * v * t  (if d kilometers, v miles/hour, t hour) Units can change form of equation

The choice of units determines the form of the equation.  It is not sufficient just to say that distance equals the velocity (speed) times the time.  We need to also specify the units of each quantity in the equation.

4.10

#### Activities and Explorations Activities:

 Play “Unit Scavenger Hunt” game Use a Meaning Table to list as many unified quantities as you can find in a particular target area (the room your in, a kitchen, a store, etc.) without repeating any units.   Compile Lists of Measuring Instruments Have teams create a list of measuring instruments.  They can do this by browsing through the Internet or retail stores making a list of measuring instruments and the scales encountered.

Explorations:

 Review Concepts of Units http://www.ex.ac.uk/cimt/dictunit/dictunit.htm (Excellent summary of units) http://metre.info/ (Extensive Details of SI or Modern Metric System) http://www.unc.edu/~rowlett/units/dictA.html  (Dictionary of Units) http://physics.nist.gov/cuu/Units/units.html  (National Institute of Standards and Technology) http://www.metricmethods.com/metricmoments.html (Common examples of Metric System)   Unit Conversions Applications http://www.chezcrowe.com/conv_app.htm (Conversion) http://www.themeter.net/  (International)

Exercises:

 Do Related Problems In Your Textbook

Copyright © 2004  Dr. Ranel E. Erickson