## A-3: UNIT SYSTEMS
Sciences are built upon measurements. Measurements are expressed with numbers. This allows the logic, precision and power of mathematics to be brought to bear on our study of nature.
As early as 1670, European scientists were recommending reform of the chaotic unit systems then in use: systems which differed from country to country. The reformers urged (l) uniformity and universality, (2) simple ratios of sizes of units, (3) rational relations between units, and (4) units referenced to constants of nature (such as the circumference of the earth, boiling point of water, etc.) In 1791, in the aftermath of the Revolution, the French National Assembly adopted a more rational system based upon decimal ratios. This came to be known as the metric system. In the United States, at this time, there was also interest in reform of units and standards. In 1786 Congress approved a decimal system of coinage. In 1790 Congress considered a report on units which Secretary of State Thomas Jefferson had prepared at the urging of George Washington. In the report Jefferson proposed, as one alternative, a decimal system of weights and measures. His system had several unfortunate features, (1) it retained some of the old unit names (pound, foot, inch, furlong, mile, etc.) but assigned them new sizes (1 foot contained 10 inches, for example), and (2) his system was not fully compatible with the metric system then being developed in France. Congress, confused and ill-informed (as usual) took no action on the proposal. John Quincy Adams' 1821 So our best opportunity for adoption of a sensible unit system slipped by. While other countries, one by one, adopted the metric system, the U. S. arrogantly went its own way, feeling no need nor desire to adopt a "foreign" system of units. In 1890, metric units were established as the legal basis of all weights and measures in
the United States, but this did little to establish the But change is coming—"slowly. Several states have marked their highway distance signs in both miles and kilometers. Some radio stations report daily temperatures in both degrees Fahrenheit and Celsius. U. S. automakers use metric parts in automobile engines. Science and medicine have been almost exclusively metric in all countries for many years. Other industries use metric standards because of international trade and competition. The relative simplicity of the metric system is well illustrated by comparing measurements of length in the United States System with those of the metric system. U. S. SYSTEM METRIC SYSTEM 1 mile = 5280 feet 1 kilometer = 1000 meter 1 mile = 1760 yards 1 hectometer = 100 meter 1 rod = 5.5 yards 1 dekameter = 10 meters 1 yard = 3 feet 1 decimeter = 1/10 meter 1 foot = 12 inches 1 centimeter = 1/100 meter 1 millimeter = 1/1000 meter 1 gallon = 4 quarts 1 liter = 1000 milliliters 1 quart = 32 fluid ounces 1 fluid ounce = 8 drams The sizes of U. S. units usually have no simple relation to each other, and the relations lack consistency. They are therefore difficult to remember. The metric units are consistently constructed and consistently named. The metric system is based upon multiples and submult- iples of 10 (very convenient for calculation in our decimal system). Also, the unit names have prefixes which consistently indicate which multiple of 10 is meant. The prefixes keep the same meanings throughout the metric system. The table on the next page shows this simplicity and consistency which is the strength of the metric system. Some of the prefixes may already he familiar. "Deci" appears in our word "decimal." "Centi" appears in our monetary system as "cent," the hundredth part of a dollar. In tax assessments one encounters the "mill" which is 1/10 cent, or a thousandth of a dollar. The prefixes even appear in slang words, as in "megabucks" for a large amount of money. Each different kind of measurement has a root name, from which other names may be constructed by combining a root name with a metric prefix. Three root names are commonly used in introductory physics courses: Names for other sizes of these three types of measurement are obtained by applying the metric prefixes. The starred names below are the only ones commonly used in physics. kilometer* kilogram* kiloliter hectometer hectogram hectoliter dekameter dekagram dekaliter meter* gram* liter* decimeter decigram deciliter centimeter* centigram centiliter millimeter milligram milliliter*
EXPONENTIAL PREFIX SYMBOL MEANING NOTATION Exa E one billion billion lO * Most commonly encountered prefixes; learn these first. The liter is really an unnecessary unit, since volumes may be expressed in cubic length units. But the liter is so entrenched in common usage, that it will probably persist for many years. One liter equals 1000.028 cubic centimeters, but in all but the most precise work the approximate relation, 1 liter = 1000 cubic centimeters is accurate enough. To give a feeling for the sizes of the metric units, we list some
1 meter = 1.09 yards
In the United States we have long used the spelling "meter" and "liter." In Europe these are spelled "metre" and "litre." There is now a lively argument over whether we should conform to the European spelling, or whether these spelling variants can be allowed, since they are unlikely to cause confusion. Most U. S. texts still prefer the "-er" form. Watch for future developments.
Measurable physical quantities (measurables) are of two kinds: (1) BASIC (or FUNDAMENTAL) MEASURABLES. Those so directly connected with measurement that they are defined by specifying a measurement process. (2) DEFINED MEASURABLES. These are defined by mathematical equations in terms of other previously defined and/or fundamental measurables. The measurables of the physical sciences are interwoven into a tight net of equations and
definitions. Remarkably few are so fundamental that they cannot be defined in terms of others.
Those few, the basic measurables, are defined through Only three basic measurables are required in mechanics. Length and time are always chosen as basic. Some unit systems choose force as the third basic measurable, others choose mass instead. But as physics developed beyond mechanics, other basic measurables were added to the list; temperature, electric current, the mole, and the luminous intensity of light are often chosen. The basic measurables are called the A few examples may clarify this. In elementary mechanics we take length, mass and time
as the dimensions of the system. For brevity these are abbreviated L, M, and T. The
dimensions of other measurables can be determined by looking at their defining equations.
Velocity is defined as the quotient of length and time; the dimensions of velocity are therefore
L/T, usually written LT Force is defined as the product of mass and acceleration; its dimensions are therefore
MLT The
As we construct equations involving measurables we discover that our choice of units can influence the form of an equation. Consider this empirical equation from engineering. d = 67.39 Here d represents the distance in feet at which a road sign is generally legible to an automobile driver, h is the height of the lettering in inches. Clearly this equation requires a statement of what units are used for b and h. If d were in miles and h in feet, the equation simply would not give physically correct answers! However, it could work for a different choice of units of we were to modify the values of the constants. d = K
By finding suitable values of the constants K To avoid such nuisances we agree upon a "standard" set of units. This allows us to write just one equation for each physical law. Such a standard set of units is called a When a coherent system is built from a basic unit set which includes mass, the system
is called A number of coherent systems have been used in the past, and are still used to some
extent. A few decades ago the cgs ( Fortunately these three systems (MKS, cgs, and FPS) are all mutually coherent for most branches of physics, especially mechanics (but not including electricity and magnetism). In mechanics the equations have the same form in all three systems, but one must be careful to express all constants (like Newton's gravitational constant) with a value appropriate to the system being used. If all equations were written with a sufficient number of constants, tables of values of these constants could be constructed for any system of units. The constants would generally have units, and might have dimensions as well.
Data and measurements may be expressed in any units, usually chosen for convenience
of size. But when this data is used in Unit conversions begin with equations which relate sizes of units, for example 1 yard = 3 feet is obviously correct, and so simple in appearance that the reader is likely to dismiss it as trivial. But such physical equations relating measurables are significantly different from equations relating pure numbers. The above equation tells us that the Equations relating measurements are manipulated by the ordinary rules of algebra, and the units are carried along according to the same rules. For example, if both sides of Eq. (3) are divided by 1 yard, the result is: Therefore: 1 = 3 feet/yard
This last expression represents an identity relation for measurements. We call it a
2.5 yards = 2.5 yards (3 feet/yard) = 7.5 feet The unit "yard" has canceled by the rules of algebra, leaving a result in feet.
velocity = 1.1 inches/second but we want an answer in feet per second. The conversion equation between feet and inches is 1 foot = 12 inches. Dividing both sides by 1 foot gives: But we actually need its reciprocal: 1 = (1/12) feet/inch (The distinction between "foot" and "feet" is grammatical, of no concern in these calculations). Now multiply our measurement by the conversion factor:
Conversion factors may be inserted as a In the physical sciences, nearly all equations are constructed in such a manner that the
When a physical equation is expressed with all quantities in one coherent unit system, the
1. An equation in which the units balance on both sides of the equal sign is called
2. An equation in which the dimensions balance on both sides of the equal sign is called
3. A unit system constructed so that all physical laws are represented by coherent
equations is called a 4. Conversion factors are homogeneous, but may be incoherent. Their primary use is to transform equations from one coherent unit set to another. 5. Such "relations" as "one cubic centimeter of water = one gram" are inhomogeneous
and incoherent, for they relate units of different dimensions and different physical meaning. An
"equality" relation such as this represents a more relaxed meaning of "equal" than is normally
permissible, though if one is careful such "conversion equations" can be used to get correct
results. Such "equations" are, however, conceptually misleading, and are best avoided at the
introductory level of physics. It is better to say "one cubic centimeter of water The tables below directly compare the commonly encountered coherent systems of units. The table of electrical units also includes conversion factors.
STANDARD ABBREVIATIONS FOR UNIT NAMES: s = second N = newton V = volt cm = centimeter lb = pound W = ohm m = meter J = joule W = watt ft = foot Hz = hertz A = Ampere g = gram mi = mile C = coulomb kg = kilogram Note 1. Unit names derived from the names of persons are
In going from right to left, use the reciprocal of the conversion factor. So, in converting amperes to abamperes, divide by 10. A column of esu/mks conversion factors is included for convenience. These entries are simply the product of the reciprocals of the other two conversion factor column entries. NOTE: c is 2.9979 x 10
1. Dimensions combine by the ordinary rules of algebra. Units do also. 2. Terms which are added or subtracted must have the same dimensions and the same units. 3. Quantities on either side of the equal sign must have the same dimensions and the same units. 4. Powers are dimensionless and unitless (though factors within them may have dimensions and units). 5. dy/dx and ∂y/∂x have the dimensions and the units of y/x (look at the formula for the definition of the derivative). 6. ∫ y dx has the dimensions and the units of yx. 7. Arguments of sin, cos, tan, log, etc. must be dimensionless, but may have units. 8. Sin, cos, tan, log, etc. are dimensionless and unitless. 9. The mathematical constants π and e are dimensionless and unitless. Test your understanding by classifying each of the following equations as homogeneous or inhomogeneous, coherent or incoherent. 1. 1 mile = 5280 feet Answers: 1. Homogeneous and incoherent. This is a correct conversion equation. 2. Homogeneous and incoherent. This equation is perfectly proper and unambiguous. 3. Inhomogeneous and incoherent (and improper!). We should avoid equating apples and oranges, unless we are interested in counting fruit! The kilogram is a unit of mass, the pound is a unit of force. We should say, "A one kilogram mass at the earth's surface would weigh 2.2 pounds."
We define some measurables in such a way that they are dimensionless. Some of these are given unit names, and some are not. Specific gravity, being a ratio of two densities, is dimensionless. It has no unit name. Index of refraction, a ratio of two speeds (of light), also has no unit name. Pi (p), the ratio of a circle's circumference to its diameter is therefore dimensionless and has no unit name. But the radian (a measure of angle) is defined as the ratio of arc length to radius, so its dimensions are L/L, that is, it is dimensionless. Yet it is given a name, the radian. The name helps to distinguish this measure from other angular measurements such as "degrees of arc" (which are also dimensionless). Sometimes measurables of physically It is also possible for different quantities with different unit names to have the same
dimensions. The quantity
(1) Bridgman.P. W. (2) Chertov, A. G. (3) Dimone, Daniel V, and Treat. (4) Isaacson, E. de St. Q, and Isaacson. (5) Jefimenko, Oleg D. (6) Scott, William Taussig. (7) Reitz, John R., Milford and Christy. (8) Dale Corson and Paul Lorrain.
(9) Symbols, Units, and Nomenclature in Physics, Document U. I. P. 11 (S. U. N. 65-3)
International Union of Pure and Applied Physics. (Reprinted in the (10) Pankhurst, R. C. (11) Perry, John.
Supply the appropriate conversion factors in the square brackets. Find the numeric value to complete the right side of the equation. The answers are on the last page. Example: Problem: 5 yards [ ] = [ ] ft. Answer: 5 yards [3 ft/yd] = 15 ft. PROBLEMS: C-1. 9 ft [ ] = [ ] yds C-2. 6 dm [ ] = [ ] cm C-3. 7 lb [ ] = [ ] cm C-4. 4 oz [ ] = [ ] lb C-5. 3 minutes [ ] = [ ] seconds C-6. 1 hour [ ] [ ] = [ ] seconds C-7. 6 seconds [ ] = [ ] hour C-8. 1 day [ ] [ ] = [ ] sec C-9. 1 year [ ] [ ] [ ] = [ ] sec C-10. 7.896 m [ ] = [ ] cm C-11. 3.428 km [ ] = [ ] cm C-12. 8.529 km [ ] = [ ] ft C-13. 20 miles/hr [ ] = [ ] miles/sec C-14. 5 cm/sec [ ] = [ ] m/s C-15. 1 mile = 1.61 km (approx), therefore: 1 mile/hr [ ] = [ ] km/hr C-16. 1 mile = 5280 ft, therefore 1 mile/hr [ ] [ ] = ft/s C-17. 1 mile/hr [ ] [ ] = m/s C-18. 1 cm C-19. 55.89 m C-20. 1 km
C-1. 9 ft [1/3 yd/ft] = [3] yds C-2. 6 dm [100 mm/dm] = [600] mm C-8. 1 yr [365 day/yr][24 hr/day][60 min/hr][60 sec/min] = 31,536,000 seconds/yr. Some students use C-10. 7.896 m [100 cm/m] = [789.6] cm C-13. 20 miles/hr [(1/3600) hr/sec] = [1/180] miles/sec © 1996, 2004 by Donald E. Simanek. |