MECHANICS GUIDE physics in introduction PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Wed, 05 Jan 2011 14:19:25 UTC Contents Articles mechanics 1 International System of Units 1 Physical quantity 9 Dimensional analysis 12 Physics 26 Scientific notation 37 Kinematics 41 Force 53 Mechanical equilibrium 75 Newton's laws of motion 77 Weight 85 Hooke's law 93 Spring (device) 102 Tension (physics) 108 Normal force 109 Rope 111 Pulley 118 Inclined plane 123 Lever 126 Wedge (mechanical device) 127 Wheel and axle 129 Momentum 131 Friction 142 Drag (physics) 151 Centripetal force 158 References Article Sources and Contributors 171 Image Sources, Licenses and Contributors 177 Article Licenses License 180 1 mechanics International System of Units The International System of Units [2] (abbreviated SI from the French Systme international d'units[3] ) is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. It is the world's most widely used system of measurement, both in everyday commerce and in science.[4] [5]

The older metric system included several groups of units. The SI was developed in 1960 from the old metre-kilogram-second system, rather than the centimetre-gram-second system, which, in turn, had a few variants. Because the SI is not static, units are created and definitions are modified through international agreement among many nations as the technology of measurement progresses, and as the precision of measurements improves. The system has been nearly globally adopted. Three principal exceptions are Burma (Myanmar), Liberia, and the United States. The United Kingdom has officially adopted the International System of Units but not with the intention of replacing customary measures entirely. Cover of brochure The International System [1] of Units . History The metric system was conceived by a group of scientists (among them, Antoine-Laurent Lavoisier, who is known as the "father of modern chemistry") who had been commissioned by the assemblee nationale and Louis XVI of France to create a unified and rational system of measures.[6] On 1 August 1793, the National Convention adopted the new decimal metre with a provisional length as well as the other decimal units with preliminary definitions and terms. On 7 April 1795 (Loi du 18 germinal, an III) the terms gramme and kilogramme replaced the former terms gravet (correctly milligrave) and grave. On 10 December 1799 (a month after Napoleon's coup d'tat), the metric system was definitively adopted in France. The desire for international cooperation on metrology led to the signing in 1875 of the Metre Convention, a treaty which established three international organizations to oversee the keeping of metric standards: General Conference on Weights and Measures (Confrence gnrale des poids et mesures or CGPM) - a meeting every four to six years of delegates from all member states; Countries by date of metrication International System of Units 2 International Bureau of Weights and Measures (Bureau international des poids et mesures or BIPM) - an international metrology centre at Svres in France; and International Committee for Weights and Measures (Comit international des poids et mesures or CIPM) - an administrative committee which meets annually at the BIPM. The history of the metric system has seen a number of variations, whose use has spread around the

world, to replace many traditional measurement systems. At the end of World War II a number of different systems of measurement were still in use throughout the world. Some of these systems were metric-system variations, whereas others were based on customary systems. It was recognised that additional steps were needed to promote a worldwide measurement system. As a result the 9th General Conference on Weights and Measures (CGPM), in 1948, asked the International Committee for Weights and Measures (CIPM) to conduct an international study of the measurement needs of the scientific, technical, and educational communities. Based on the findings of this study, the 10th CGPM in 1954 decided that an international system should be derived from six base units to provide for the measurement of temperature and optical radiation in addition to mechanical and electromagnetic quantities. The six base units that were recommended are the metre, kilogram, second, ampere, degree Kelvin (later renamed the kelvin), and the candela. In 1960, the 11th CGPM named the system the International System of Units, abbreviated SI from the French name: Le Systme international d'units. The seventh base unit, the mole, was added in 1971 by the 14th CGPM. One of the CIPM committees, the CCU, has proposed a number of changes to the definitions of the base units used in SI. It is expected that the CGPM will consider this proposal in October 2011.[7] Related systems The definitions of the concepts 'quantity', 'unit', 'dimension' etc. used in measurement, are given in the International Vocabulary of Metrology.[8] The quantities and equations which define the SI units are now referred to as the International System of Quantities (ISQ), and are set out in the ISO/IEC 80000 Quantities and Units. A readable discussion of the present units and standards is found at Brian W. Petley [9] International Union of Pure and Applied Physics I.U.P.A.P.- 39 (2004). Units The International System of Units consists of a set of units together with a set of prefixes. The units are divided into two classesbase units and derived units. There are seven base units, each representing, by convention, different kinds of physical quantities. SI base units[10] [11] Name Unit symbol metre m kilogram kg Quantity Symbol length l (a lowercase L) mass m second s time t ampere A electric current I (a capital i)

kelvin K thermodynamic temperature T candela cd luminous intensity Iv (a capital i with lowercase v subscript) mole mol amount of substance n International System of Units 3 There are an unlimited number of derived units formed from multiplication and division of the seven base units,[12] for example the SI derived unit of speed is metre per second, m/s. Some derived units have special names; for example, the unit of resistance, the ohm, symbol , is uniquely defined by the relation = m2kgs3A2, which follows from the definition of the quantity electrical resistance. The radian and steradian, once given special status, are now considered derived units.[12] A prefix may be added to a unit to produce a multiple of the original unit. All multiples are integer powers of ten, and beyond a hundred(th) all are integer powers of a thousand. For example, kilo- denotes a multiple of a thousand and milli- denotes a multiple of a thousandth; hence there are one thousand millimetres to the metre and one thousand metres to the kilometre. The prefixes are never combined: a millionth of a kilogram is a milligram not a microkilogram. Standard prefixes for the SI units of measure Multiples Name deca- hecto- kilo- mega- giga- tera- peta- exa- zetta- yottaSymbol da Factor Fractions 100 101 hkMGTPEZY 102 103 106 109 1012 1015 1018 1021 1024 Name deci- centi- milli- micro- nano- pico- femto- atto- zepto- yoctoSymbol d Factor 100 101 cmnp f 102 103 106 109 1012 1015 a z 1018 1021 y 1024 In addition to the SI units, there is also a set of non-SI units accepted for use with SI which includes some commonly used non-coherent units such as the litre. Writing unit symbols and the values of quantities The value of a quantity is written as a number followed by a space (representing a multiplication sign) and a unit symbol; e.g., "2.21 kg", "7.3 102 m2", "22 K". This rule explicitly includes the percent sign (%). Exceptions are

the symbols for plane angular degrees, minutes and seconds (, and ), which are placed immediately after the number with no intervening space.[13] [14] Symbols for derived units formed by multiplication are joined with a centre dot () or a non-break space, for example, "Nm" or "N m". Symbols for derived units formed by division are joined with a solidus (), or given as a negative exponent. For example, the "metre per second" can be written "ms", "m s1", "ms1" or . Only one solidus should be used; e.g., "kg(ms2)" or "kgm1s2" are acceptable but "kgms2" is ambiguous and unacceptable. Many computer users will type the / character provided on computer keyboards, which in turn produces the Unicode character U+002F, which is named solidus but is distinct from the Unicode solidus character, U+2044. Symbols are mathematical entities, not abbreviations, and do not have an appended period/full stop (.). Symbols are written in upright (Roman) type (m for metres, s for seconds), so as to differentiate from the italic type used for quantities (m for mass, s for displacement). By consensus of international standards bodies, this rule is applied independent of the font used for surrounding text.[15] Symbols for units are written in lower case (e.g., "m", "s", "mol"), except for symbols derived from the name of a person (e.g., "Pa", "Hz", "K" for Pascal, Hertz, Kelvin).[16] The one exception is the litre, whose original symbol "l" is unsuitably similar to the numeral "1" or the uppercase letter "i" (depending on the typeface used), at least in many English-speaking countries. The American National Institute of Standards and Technology recommends that "L" be used instead, a usage which is common in the US, Canada and Australia (but not elsewhere). This has been accepted as an alternative by International System of Units the CGPM since 1979. The cursive l is occasionally seen, especially in Japan and Greece, but this is not currently recommended by any standards body. For more information, see litre. A prefix is part of the unit, and its symbol is prepended to the unit symbol without a separator (e.g., "k" in "km", "M" in "MPa", "G" in "GHz" and so on). Compound prefixes are not allowed. Symbols of units are not pluralised, for example "25 kg" (not "25 kgs").[15] The 10th resolution of CGPM in 2003 declared that "the symbol for the decimal marker shall be either the point on the line or the comma on the line." In practice, the decimal point is used in English-speaking countries as well as most of Asia and the comma in most continental European languages.

Spaces may be used as a thousands separator (1000000) in contrast to commas or periods (1,000,000 or 1.000.000) in order to reduce confusion resulting from the variation between these forms in different countries. In print, the space used for this purpose is typically narrower than that between words (commonly a thin space). Any line-break inside a number, inside a compound unit, or between number and unit should be avoided, but, if necessary, the last-named option should be used. In Chinese, Japanese, and Korean language computing (CJK), some of the commonly used units, prefix-unit combinations, or unit-exponent combinations have been allocated predefined single characters taking up a full square. Unicode includes these in its CJK Compatibility [17] and Letterlike Symbols [18] subranges for back compatibility, without necessarily recommending future usage. When writing dimensionless quantities, the terms 'ppb' (parts per billion) and 'ppt' (parts per trillion) are recognised as language-dependent terms, since the value of billion and trillion can vary from language to language. SI, therefore, recommends avoiding these terms.[19] However, no alternative is suggested by the International Bureau of Weights and Measures (BIPM). Writing the unit names Names of units start with a lower-case letter, even when the symbol for the unit begins with a capital letter (e.g., newton, hertz, pascal). This also applies to 'degrees Celsius', since 'degree' is the unit. Names of units are pluralised using the normal English grammar rules,[20] [21] for example, "henries" is the plural of "henry".[20] :31 The units lux, hertz, and siemens are exceptions from this rule: they remain the same in singular and plural. The official US spellings for deca, metre, and litre are deka, meter, and liter, respectively.[22] Realisation of units Metrologists carefully distinguish between the definition of a unit and its realisation. The definition of each base unit of the SI is drawn up so that it is unique and provides a sound theoretical basis upon which the most accurate and reproducible measurements can be made. The realisation of the definition of a unit is the procedure by which the definition may be used to establish the value and associated uncertainty of a quantity of the same kind as the unit. A description of how the definitions of some important units are realised in practice is given on the BIPM website.[23] However, "any method consistent with the laws of physics could be used to realise any SI unit."[24] (p. 111). 4 International System of Units 5 Conversion factors The relationship between the units used in different systems is determined by convention or from the basic definition

of the units. Conversion of units from one system to another is accomplished by use of a conversion factor. There are several compilations of conversion factors; see, for example, Appendix B of NIST SP 811.[20] Cultural issues The near-worldwide adoption of the metric system as a tool of economy and everyday commerce was based to some extent on the lack of customary systems in many countries to adequately describe some concepts, or as a result of an attempt to standardise the many regional variations in the customary system. International factors also affected the adoption of the metric system, as many countries increased their trade. For use in science, it simplifies dealing with very large and small quantities, since it lines up so well with the decimal numeral system. Three nations have not officially adopted the International System of Units as their primary or sole system of measurement: Myanmar (Burma), Liberia, and the United States Many units in everyday and scientific use are not derived from the seven SI base units (metre, kilogram, second, ampere, kelvin, mole, and candela) combined with the SI prefixes. In some cases these deviations have been approved by the BIPM.[25] Some examples include: The many units of time (minute, min; hour, h; day, d) in use besides the SI second, and are specifically accepted for use according to table 6.[26] The year is specifically not included but has a recommended conversion factor.[27] The Celsius temperature scale; kelvins are rarely employed in everyday use. Electric energy is often billed in kilowatt-hours instead of megajoules. Similarly, battery charge is often measured as milliamperes-hour (mAh) instead of coulombs. The nautical mile and knot (nautical mile per hour) used to measure travel distance and speed of ships and aircraft (1 International nautical mile = 1852 m or approximately 1 minute of latitude). In addition to these, Annex 5 of the Convention on International Civil Aviation permits the "temporary use" of the foot for altitude. Astronomical distances measured in astronomical units, parsecs, and light-years instead of, for example, petametres (a light-year is about 9.461 Pm or about 9461000000000000 m). Atomic scale units used in physics and chemistry, such as the ngstrm, electron volt, atomic mass unit and barn. Some physicists prefer the centimetre-gram-second (CGS) units, or systems based on physical constants, such as Planck units, atomic units, or geometric units. In some countries, the informal cup measurement has become 250 mL. Likewise, a 500 g metric pound is used in many countries. Liquids, especially alcoholic ones, are often sold in units whose origins are historical (for example, pints for beer and cider in glasses in the UK although pint means 568 mL; champagne in Jeroboams in France).

A metric mile of 10 km is used in Norway and Sweden. The term metric mile is also used in some English speaking countries for the 1500 m foot race. In the US, blood glucose measurements are recorded in milligrams per decilitre (mg/dL), which would normalise to cg/L; in Canada, Australia, New Zealand, Oceania, and Europe, the standard is millimole per litre (mmol/L) or mM (millimolar). Blood pressure and atmospheric pressure are usually measured in mmHg and bars, respectively, instead of Pa. International System of Units The fine-tuning that has happened to the metric base-unit definitions over the past 200 years, as experts have tried periodically to find more precise and reproducible methods, does not affect the everyday use of metric units. Since most non-SI units in common use, such as the US customary units, are defined in SI units,[28] any change in the definition of the SI units results in a change of the definition of the older units, as well. International trade One of the European Union's (EU) objectives is the creation of a single market for trade. In order to achieve this objective, the EU standardised on using SI as the legal units of measure. At the time of writing (2009) it had issued two units of measurement directives which catalogued the units of measure that might be used for, amongst other things, trade: the first was Directive 71/354/EEC[29] issued in 1971 which required member states to standardise on SI rather than use the variety of cgs and mks units then in use. The second was Directive 80/181/EEC[30] [31] [32] [33] [34] issued in 1979 which replaced the first and which gave the United Kingdom and the Republic of Ireland a number of derogations from the original directive. The directives gave a derogation from using SI units in areas where other units of measure had either been agreed by international treaty or which were in universal use in worldwide trade. They also permitted the use of supplementary indicators alongside, but not in place of the units catalogued in the directive. In its original form, Directive 80/181/EEC had a cut-off date for the use of such indicators, but with each amendment this date was moved until, in 2009, supplementary indicators have been allowed indefinitely. References [1] http:/ / www. bipm. org/ en/ publications/ brochure/ [2] International Bureau of Weights and Measures (2006), The International System of Units (SI) (http:/ / www. bipm. org/ utils/ common/ pdf/ si_brochure_8_en. pdf) (8th ed.), ISBN 92-822-2213-6, [3] Resolution of the International Bureau of Weights and Measures establishing the International System of Units (http:/ / www. bipm. org/ en/ CGPM/ db/ 11/ 12/ ) [4] Official BIPM definitions (http:/ / www. bipm. org/ en/ si/ base_units/ ) [5] An extensive presentation of the SI units is maintained on line by NIST (http:/ / www. physics.

nist. gov/ cuu/ Units/ units. html), including a diagram (http:/ / www. physics. nist. gov/ cuu/ Units/ SIdiagram. html) of the interrelations between the derived units based upon the SI units. Definitions of the basic units can be found on this site, as well as the CODATA report (http:/ / physics. nist. gov/ cuu/ Constants/ codata. pdf) listing values for special constants such as the electric constant, the magnetic constant and the speed of light, all of which have defined values as a result of the definition of the metre and ampere. In the International System of Units (SI) (BIPM, 2006), the definition of the meter fixes the speed of light in vacuum c0, the definition of the ampere fixes the magnetic constant (also called the permeability of vacuum) 0, and the definition of the mole fixes the molar mass of the carbon 12 atom M(12C) to have the exact values given in the table [Table 1, p.7]. Since the electric constant (also called the permittivity of vacuum) is related to 0 by 0 = 1/0c02, it too is known exactly. CODATA report [6] "The name "kilogram"" (http:/ / www1. bipm. org/ en/ si/ history-si/ name_kg. html). . Retrieved 25 July 2006. [7] Ian Mills (29 September 2010). "Draft Chapter 2 for SI Brochure, following redefinitions of the base units" (http:/ / www. bipm. org/ utils/ en/ pdf/ si_brochure_draft_ch2. pdf). CCU. . Retrieved 2011-01-01. [8] "The International Vocabulary of Metrology (VIM)" (http:/ / www. bipm. org/ en/ publications/ guides/ vim. html). . [9] http:/ / www. physics. ohio-state. edu/ ~jossem/ IUPAP/ PhysicsNowText-A4-1. pdf [10] Barry N. Taylor & Ambler Thompson Ed. (2008). The International System of Units (SI) (http:/ / physics. nist. gov/ Pubs/ SP330/ sp330. pdf). Gaithersburg, MD: National Institute of Standards and Technology. pp. 23. . Retrieved 18 June 2008. [11] Quantities Units and Symbols in Physical Chemistry (http:/ / old. iupac. org/ publications/ books/ author/ mills. html), IUPAC [12] Ambler Thompson and Barry N. Taylor, (2008), Guide for the Use of the International System of Units (SI) (http:/ / physics. nist. gov/ cuu/ pdf/ sp811. pdf), (Special publication 811), Gaithersburg, MD: National Institute of Standards and Technology, p. 3, footnote 2. [13] The International System of Units (SI) (http:/ / www. bipm. org/ utils/ common/ pdf/ si_brochure_8_en. pdf) (8 ed.). International Bureau of Weights and Measures (BIPM). 2006. p. 133. . [14] Thompson, A.; Taylor, B. N. (July 2008). "NIST Guide to SI Units Rules and Style Conventions" (http:/ / physics. nist. gov/ Pubs/ SP811/ sec07. html). National Institute of Standards and Technology. . Retrieved 29 December 2009. 6 International System of Units [15] Bureau International des Poids et Mesures (2006). The International System of Units (SI) (http:/ / www. bipm. org/ utils/ common/ pdf/ si_brochure_8_en. pdf). 8th ed.. . Retrieved 13 February 2008. Chapter 5. [16] Ambler Thompson and Barry N. Taylor, (2008), Guide for the Use of the International System of Units (SI) (http:/ / physics. nist. gov/ cuu/ pdf/ sp811. pdf), (Special publication 811), Gaithersburg, MD: National Institute of Standards and Technology, section 6.1.2 [17] http:/ / www. unicode. org/ charts/ PDF/ U3300. pdf

[18] http:/ / www. unicode. org/ charts/ PDF/ U2100. pdf [19] http:/ / www. bipm. org/ en/ si/ si_brochure/ chapter5/ 5-3-7. html [20] Ambler Thompson & Barry N. Taylor (2008). NIST Special Publication 811: Guide for the Use of the International System of Units (SI) (http:/ / physics. nist. gov/ cuu/ pdf/ sp811. pdf). National Institute of Standards and Technology. . Retrieved 18 June 2008. [21] Turner, James M. (9 May 2008). May 2008/pdf/E8-11058.pdf "Interpretation of the International System of Units (the Metric System of Measurement) for the United States" (http:/ / www. gpo. gov/ fdsys/ pkg/ FR-16). Federal Register (National Archives and Records Administration) 73 (96): 284323. FR Doc number E8-11058. May 2008/pdf/E8-11058.pdf. Retrieved 28 October 2009. [22] "The International System of Units" (http:/ / physics. nist. gov/ Pubs/ SP330/ sp330. pdf). pp. iii. . Retrieved 27 May 2008. [23] SI Practical Realization brochure (http:/ / www. bipm. org/ en/ si/ si_brochure/ appendix2/ ) [24] International Bureau of Weights and Measures (2006), The International System of Units (SI) (http:/ / www. bipm. org/ utils/ common/ pdf/ si_brochure_8_en. pdf) (8th ed.), p. 111, ISBN 92-822-2213-6, [25] BIPM - Table 8 (http:/ / www. bipm. org/ en/ si/ si_brochure/ chapter4/ table8. html) [26] BIPM - Table 6 (http:/ / www. bipm. org/ en/ si/ si_brochure/ chapter4/ table6. html) [27] NIST Guide to SI Units - Appendix B9. Conversion Factors (http:/ / physics. nist. gov/ Pubs/ SP811/ appenB9. html#TIME) [28] Mendenhall, T. C. (1893). "Fundamental Standards of Length and Mass". Reprinted in Barbrow, Louis E. and Judson, Lewis V. (1976). Weights and measures standards of the United States: A brief history (NBS Special Publication 447). Washington D.C.: Superintendent of Documents. Viewed 23 August 2006 at (http:/ / physics. nist. gov/ Pubs/ SP447/ ) pp. 2829. [29] "Council Directive of 18 October 1971 on the approximation of laws of the member states relating to units of measurement, (71/354/EEC)" (http:/ / eur-lex. europa. eu/ Notice. do?mode=dbl& lang=en& lng1=en,nl& lng2=da,de,el,en,es,fr,it,nl,pt,& val=22924:cs& page=1& hwords=). . Retrieved 7 February 2009. [30] The Council of the European Communities (21 December 1979). "Council Directive 80/181/EEC of 20 December 1979 on the approximation of the laws of the Member States relating to Unit of measurement and on the repeal of Directive 71/354/EEC" (http:/ / eur-lex. europa. eu/ LexUriServ/ LexUriServ. do?uri=CONSLEG:1980L0181:19791221:EN:PDF). . Retrieved 7 February 2009. [31] The Council of the European Communities (20 December 1984). "Council Directive 80/181/EEC of 20 December 1979 on the approximation of the laws of the Member States relating to Unit of measurement and on the repeal of Directive 71/354/EEC" (http:/ / eur-lex. europa. eu/ LexUriServ/ LexUriServ. do?uri=CONSLEG:1980L0181:19841220:EN:PDF). . Retrieved 7 February 2009. [32] The Council of the European Communities (30 November 1989). "Council Directive 80/181/EEC of 20 December 1979 on the approximation of the laws of the Member States relating to Unit of measurement and on the repeal of Directive 71/354/EEC" (http:/ / eur-lex. europa. eu/ LexUriServ/ LexUriServ. do?uri=CONSLEG:1980L0181:19891130:EN:PDF). . Retrieved 7 February 2009. [33] The Council of the European Communities (9 February 2000). "Council Directive 80/181/EEC of 20 December 1979 on the approximation

of the laws of the Member States relating to Unit of measurement and on the repeal of Directive 71/354/EEC" (http:/ / eur-lex. europa. eu/ LexUriServ/ LexUriServ. do?uri=CONSLEG:1980L0181:20000209:EN:PDF). . Retrieved 7 February 2009. [34] The Council of the European Communities (27 May 2009). "Council Directive 80/181/EEC of 20 December 1979 on the approximation of the laws of the Member States relating to Unit of measurement and on the repeal of Directive 71/354/EEC" (http:/ / eur-lex. europa. eu/ LexUriServ/ LexUriServ. do?uri=CONSLEG:1980L0181:20090527:EN:PDF). . Retrieved 14 September 2009. Further reading International Union of Pure and Applied Chemistry (1993). Quantities, Units and Symbols in Physical Chemistry, 2nd edition, Oxford: Blackwell Science. ISBN 0-632-03583-8. Electronic version. (http://www.iupac.org/ publications/books/gbook/green_book_2ed.pdf) Unit Systems in Electromagnetism (http://info.ee.surrey.ac.uk/Workshop/advice/coils/unit_systems/#rms) MW Keller et al. (http://qdev.boulder.nist.gov/817.03/pubs/downloads/set/Watt_Triangle_sub1.pdf) Metrology Triangle Using a Watt Balance, a Calculable Capacitor, and a Single-Electron Tunneling Device 7 International System of Units External links Official BIPM Bureau International des Poids et Mesures (SI maintenance agency) (http://www.bipm.org/en/si/) (home page) BIPM brochure (http://www.bipm.org/en/si/si_brochure/) (SI reference) ISO 80000-1:2009 Quantities and units -- Part 1: General (http://www.iso.org/iso/iso_catalogue/ catalogue_ics/catalogue_detail_ics.htm?csnumber=30669) NIST Official Publications (http://physics.nist.gov/cuu/Units/bibliography.html) NIST Special Publication 330, 2008 Edition: The International System of Units (SI) (http://physics.nist.gov/ Pubs/SP330/sp330.pdf) NIST Special Pub 814: Interpretation of the SI for the United States and Federal Government Metric Conversion Policy (http://ts.nist.gov/WeightsAndMeasures/Metric/pub814.cfm) Weights and Measures Act, Canada (http://laws.justice.gc.ca/en/ShowTdm/cs/W-6///en) IEEE/ASTM SI 10-2002 Standard for Use of the International System of Units (SI): The Modern Metric System (http://webstore.ansi.org/ansidocstore/product.asp?sku=SI10-2002) (ANSI approved, joint IEEE/ASTM standard) Rules for SAE Use of SI (Metric) Units (http://www.sae.org/standardsdev/tsb/tsb003.pdf) National Physical Laboratory, UK (http://www.npl.co.uk/server.php?show=category.364) Information International System of Units (http://www.dmoz.org/Science/Reference/Units_of_Measurement//) at the Open Directory Project EngNet Metric Conversion Chart (http://www.engnetglobal.com/tips/convert.aspx) Online

Categorised Metric Conversion Calculator U.S. Metric Association. 2008. A Practical Guide to the International System of Units (http://lamar.colostate. edu/~hillger/pdf/Practical_Guide_to_the_SI.pdf) History LaTeX SIunits package manual (ftp://cam.ctan.org/texarchive/macros/latex/contrib/SIunits/SIunits.pdf) gives a historical background to the SI system. Research The metrological triangle (http://www.npl.co.uk/server.php?show=ConWebDoc.1835) Recommendation of ICWM 1 (CI-2005) (http://www.bipm.org/cc/CIPM/Allowed/94/ CIPM-Recom1CI-2005-EN.pdf) Pro-metric advocacy groups The UK Metric Association (http://www.ukma.org.uk/) The US Metric Association (http://www.metric.org/) Canadian Metric Association (http://niagara.cioc.ca/details.asp?RSN=5108&Number=0) Metrication US (http://www.metrication.us) Pro-customary measures pressure groups Pro-customary measures groups (http://www.dmoz.org/Society/Issues/Government_Operations/ Anti-Metrication//) at the Open Directory Project 8 Physical quantity 9 Physical quantity A physical quantity is a physical property that can be quantified by measurement. Formally, the International Vocabulary of Metrology', 3rd edition (VIM) defines quantity as: property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed as a number and a reference[1] Hence the value of a physical quantity Q is expressed as the product of a numerical value {Q} and a unit of measurement [Q]. Q = {Q} x [Q] Quantity calculus describes how to do math with quantities. Examples If the temperature T of a body is quantified (measured) as 300 degrees Kelvin this is written T = 300 x K = 300 K where T is the symbol of the physical quantity (NB degrees Celsius cannot be treated in this way) If a person weighs 120 pounds, then "120" is the numerical value and "pound" is the unit. This physical quantity mass would be written as "120 lbs", or m = 120 lbs An example employing SI units and scientific notation for the number, might be a measurement of power written as P = 42.3 x 103 W, Here, P represents the physical quantity of power, 42.3 x 103 is the numerical value {P}, and W is the symbol for the

unit of power [P], the watt Symbols for physical quantities Usually, the symbols for physical quantities are chosen to be a single letter of the Latin or Greek alphabet, and are often printed in italic type. Often, the symbols are modified by subscripts and superscripts, to specify what they refer to for instance Ek is usually used to denote kinetic energy and cp heat capacity at constant pressure. (Note the difference in the style of the subscripts: k is the abbreviation of the word kinetic, whereas p is the symbol for the physical quantity pressure rather than an abbreviation of the word "pressure".) Symbols for quantities should be chosen according to the international recommendations from ISO 31, the IUPAP red book and the IUPAC green book. For example, the recommended symbol for the physical quantity 'mass' is m, and the recommended symbol for the quantity 'charge' is Q. Symbols for physical quantities that are vectors are bold italic type. If, e.g., u is the speed of a particle, then the straightforward notation for its velocity is u. Numerical quantities, even those denoted by letters, are usually printed in Roman (upright) type, e.g.: 1, 2, e (for the base of natural logarithm), i (for the imaginary unit) or (for 3.14...). Symbols for numerical functions such as sin are Roman type too. Although the recommendation is not followed by Wikipedia, operators like d in dx are recommended also to be printed in Roman type. Physical quantity 10 Units of physical quantities Most physical quantities Q include a unit [Q] (where [Q] means "unit of Q"). Neither the name of a physical quantity, nor the symbol used to denote it, implies a particular choice of unit. For example, a quantity of mass might be represented by the symbol m, and could be expressed in the units kilograms (kg), pounds (lb), or Daltons (Da). SI units are usually preferred today. Base quantities, derived quantities and dimensions The notion of physical dimension of a physical quantity was introduced by Fourier in 1822.[2] By convention, physical quantities are organized in a dimensional system built upon base quantities, each of which is regarded as having its own dimension. The seven base quantities of the International System of Quantities (ISQ) and their corresponding SI units are listed in the following table. Other conventions may have a different number of fundamental units (e.g. the CGS and MKS systems of units). International System of Units base quantities Name Symbol for quantity Symbol for dimension SI base unit Symbol for unit Length l, x, r, etc. L meter m

Time t T second s Mass m M kilogram kg Electric current I, i I ampere A Thermodynamic temperature T kelvin K Amount of substance n N mole mol Luminous intensity Iv J candela cd All other quantities are derived quantities since their dimensions are derived from those of base quantities by multiplication and division. For example, the physical quantity velocity is derived from base quantities length and time and has dimension L/T. Some derived physical quantities have dimension 1 and are said to be dimensionless quantities. Extensive and intensive quantities A quantity is called: extensive when its magnitude is additive for subsystems (volume, mass, etc.) intensive when the magnitude is independent of the extent of the system (temperature, pressure, etc.) Some physical quantities are prefixed in order to further qualify their meaning: specific is added to refer to a quantity which is expressed per unit mass (such as specific heat capacity) molar is added to refer to a quantity which is expressed per unit amount of substance (such as molar volume) There are also physical quantities that can be classified as neither extensive nor intensive, for example angular momentum, area, force, length, and time. Physical quantity 11 Physical quantities as coordinates over spaces of physical qualities The meaning of the term physical quantity is generally well understood (everyone understands what is meant by the frequency of a periodic phenomenon, or the resistance of an electric wire). It is clear that behind a set of quantities like temperature inverse temperature logarithmic temperature, there is a qualitative notion: the coldhot quality. Over this one-dimensional quality space, we may choose different coordinates: the temperature, the inverse temperature, etc. Other quality spaces are multidimensional. For instance, to represent the properties of an ideal elastic medium we need 21 coefficients, that can be the 21 components of the elastic stiffness tensor , or the 21 components of the elastic compliance tensor (inverse of the stiffness tensor), or the proper elements (six eigenvalues and 15 angles) of any of the two tensors, etc. Again, we are selecting coordinates over a 21dimensional quality space. On this space, each point represents a particular elastic medium. It is always possible to define the distance between two points of any quality space, and this distance is inside a given theoretical context uniquely defined. For instance, two periodic phenomena can be characterized by their periods, and

, or by their frequencies, and . The only definition of distance that respects some clearly defined invariances is log log . These notions have implications in physics. As soon as we accept that behind the usual physical quantities there are quality spaces, that usual quantities are only special coordinates over these quality spaces, and that there is a metric in each space, the following question arises: Can we do physics intrinsically, i.e., can we develop physics using directly the notion of physical quality, and of metric, and without using particular coordinates (i.e., without any particular choice of physical quantities)? In fact, physics can (and must?) be developed independently of any particular choice of coordinates over the quality spaces, i.e., independently of any particular choice of physical quantities to represent the measurable physical qualities.[3] See also Physical constant Notes [1] Joint Committee for Guides in Metrology (JCGM), International Vocabulary of Metrology, Basic and General Concepts and Associated Terms (VIM), III ed., Pavillon de Breteuil : JCGM 200:2008, 1.1 ( on-line (http:/ / www. bipm. org/ utils/ common/ documents/ jcgm/ JCGM_200_2008. pdf)) [2] Fourier, Joseph. Thorie analytique de la chaleur, Firmin Didot, Paris, 1822. (In this book, Fourier introduces the concept of physical dimensions for the physical quantities.) [3] Tarantola, Albert. Elements for physics - Quantities, qualities and intrinsic theories, Springer, 2006. ISBN 3-540-25302-5. (http:/ / www. ipgp. jussieu. fr/ ~tarantola/ Files/ Professional/ Books/ ElementsForPhysics-ScreenViewing. pdf) References Cook, Alan H. The observational foundations of physics, Cambridge, 1994. ISBN 0-521-45597Dimensional analysis Dimensional analysis In physics and science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions (usually mass, length, time, electric charge, and temperature) which describe it; for example, speed has the dimension length / time, and may be measured in meters per second, miles per hour, or other units. Dimensional analysis is based on the fact that a physical law must be independent of the units used to measure the physical variables. A straightforward practical consequence is that any meaningful equation (and any inequality and inequation) must have the same dimensions in the left and right sides. Checking this is the basic way of performing dimensional analysis. Dimensional analysis is routinely used to check the plausibility of derived equations and computations. It is also

used to form reasonable hypotheses about complex physical situations that can be tested by experiment or by more developed theories of the phenomena, and to categorize types of physical quantities and units based on their relations to or dependence on other units, or their dimensions if any. The basic principle of dimensional analysis was known to Isaac Newton (1686) who referred to it as the "Great Principle of Similitude".[1] James Clerk Maxwell played a major role in establishing modern use of dimensional analysis by distinguishing mass, length, and time as fundamental units, while referring to other units as derived.[2] The 19th-century French mathematician Joseph Fourier made important contributions[3] based on the idea that physical laws like F = ma should be independent of the units employed to measure the physical variables. This led to the conclusion that meaningful laws must be homogeneous equations in their various units of measurement, a result which was eventually formalized in the Buckingham theorem. This theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n m dimensionless parameters, where m is the number of fundamental dimensions used. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables. A dimensional equation can have the dimensions reduced or eliminated through nondimensionalization, which begins with dimensional analysis, and involves scaling quantities by characteristic units of a system or natural units of nature. This gives insight into the fundamental properties of the system, as illustrated in the examples below. Introduction Definition The dimensions of a physical quantity are associated with combinations of mass, length, time, electric charge, and temperature, represented by sans-serif symbols M, L, T, Q, and , respectively, each raised to rational powers. The term dimension is more abstract than scale unit: mass is a dimension, while kilograms are a scale unit (choice of standard) in the mass dimension. As examples, the dimension of the physical quantity speed is distance/time (L/T or LT 1), and the dimension of the physical quantity force is "mass acceleration" or "mass(distance/time)/time" (ML/T2 or MLT 2). In principle, other dimensions of physical quantity could be defined as "fundamental" (such as momentum or energy or electric current) in lieu of some of those shown above. Most physicists do not recognize temperature, , as a fundamental dimension of physical quantity since it essentially expresses the energy per particle per degree of freedom, which can be expressed in terms of energy (or mass, length, and time). Still others do not recognize electric charge, Q, as a separate fundamental dimension of physical quantity, since it has been expressed in terms of mass,

length, and time in unit systems such as the cgs system. There are also physicists that have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity.[4] The unit of a physical quantity and its dimension are related, but not identical concepts. The units of a physical quantity are defined by convention and related to some standard; e.g., length may have units of meters, feet, inches, 12 Dimensional analysis miles or micrometres; but any length always has a dimension of L, independent of what units are arbitrarily chosen to measure it. Two different units of the same physical quantity have conversion factors that relate them. For example: 1 in = 2.54 cm; then (2.54 cm/in) is the conversion factor, and is itself dimensionless and equal to one. Therefore multiplying by that conversion factor does not change a quantity. Dimensional symbols do not have conversion factors. Mathematical properties Dimensional symbols, such as L, form a group: The identity is defined as L0 = 1, and the inverse to L is 1/L or L1. L raised to any rational power p is a member of the group, having an inverse of Lp or 1/Lp. The operation of the group is multiplication, having the usual rules for handling exponents (Ln Lm = Ln+m). Dimensional symbols form a vector space over the rational numbers, with for example dimensional symbol MiLjTk corresponding to the vector (i,j,k). When physical measured quantities (be they like-dimensioned or unlike-dimensioned) are multiplied or divided by one other, their dimensional units are likewise multiplied or divided; this corresponds to addition or subtraction in the vector space. When measurable quantities are raised to a rational power, the same is done to the dimensional symbols attached to those quantities; this corresponds to scalar multiplication in the vector space. A basis for a given vector space of dimensional symbols is called a set of fundamental units or fundamental dimensions, and all other vectors are called derived units. As in any vector space, one may choose different bases, which yields different systems of units (e.g., choosing whether the unit for charge is derived from the unit for current, or vice versa). Dimensionless quantities correspond to the origin in this vector space. The set of units of the physical quantities involved in a problem correspond to a set of vectors (or a matrix). The kernel describes some number (e.g., m) of ways in which these vectors can be combined to produce a zero vector. These correspond to producing (from the measurements) a number of dimensionless quantities, {1,...,m}. (In fact these ways completely span the null subspace of another different space, of powers of the measurements.) Every possible way of multiplying (and exponating) together the measured quantities to produce

something with the same units as some derived quantity X can be expressed in the general form Consequently, every possible commensurate equation for the physics of the system can be rewritten in the form . Knowing this restriction can be a powerful tool for obtaining new insight into the system. Mechanics In mechanics, the dimension of any physical quantity can be expressed in terms of the fundamental dimensions (or base dimensions) M, L, and T these form a 3-dimensional vector space. This is not the only possible choice, but it is the one most commonly used. For example, one might choose force, length and mass as the base dimensions (as some have done), with associated dimensions F, L, M; this corresponds to a different basis, and one may convert between these representations by a change of basis. The choice of the base set of dimensions is, thus, partly a convention, resulting in increased utility and familiarity. It is, however, important to note that the choice of the set of dimensions cannot be chosen arbitrarily it is not just a convention because the dimensions must form a basis: they must span the space, and be linearly independent. For example, F, L, M form a set of fundamental dimensions because they form an equivalent basis to M, L, T: the former can be expressed as [F=ML/T2],L,M while the latter can be expressed as M,L, [T=(ML/F)1/2]. On the other hand, using length, velocity and time (L, V, T) as base dimensions will not work well (they do not form a set of fundamental dimensions), for two reasons: 13 Dimensional analysis There is no way to obtain mass or anything derived from it, such as force without introducing another base dimension (thus these do not span the space). Velocity, being derived from length and time (V=L/T), is redundant (the set is not linearly independent). Other fields of physics and chemistry Depending on the field of physics, it may be advantageous to choose one or another extended set of dimensional symbols. In electromagnetism, for example, it may be useful to use dimensions of M, L, T, and Q, where Q represents quantity of electric charge. In thermodynamics, the base set of dimensions is often extended to include a dimension for temperature, . In chemistry the number of moles of substance (loosely, but not precisely, related to the number of molecules or atoms) is often involved and a dimension for this is used as well. The choice of the dimensions or even the number of dimensions to be used in different fields of physics is to some extent arbitrary, but consistency in use and ease of communications are very important. Commensurability The most basic consequence of dimensional analysis is: Only commensurable quantities (quantities with the same dimensions) may be compared, equated,

added, or subtracted. However, One may take ratios of incommensurable quantities (quantities with different dimensions), and multiply or divide them. For example, it makes no sense to ask if 1 hour is more or less than 1 kilometer, as these have different dimensions, nor to add 1 hour to 1 kilometer. On the other hand, if an object travels 100 km in 2 hours, one may divide these and conclude that the object's average speed was 50 km/hour. As a corollary of this requirement, it follows that in a physically meaningful expression, only quantities of the same dimension can be added, subtracted, or compared. For example, if mman, mrat and Lman denote, respectively, the mass of some man, the mass of a rat and the length of that man, the expression mman + mrat is meaningful, but mman + Lman is meaningless. However, mman/L2man is fine. Thus, dimensional analysis may be used as a sanity check of physical equations: the two sides of any equation must be commensurable or have the same dimensions, i.e., the equation must be dimensionally homogeneous. Even when two physical quantities have identical dimensions, it may be meaningless to compare or add them. For example, although torque and energy share the dimension ML2T2, they are fundamentally different physical quantities. To compare, add, or subtract quantities with the same dimensions but expressed in different units, the standard procedure is to first convert them all to the same units. For example, to compare 32 metres with 35 yards, use 1 yard = 0.9144 m to convert 35 yards to 32.004 m. 14 Dimensional analysis Polynomials and transcendental functions Scalar arguments to transcendental functions such as exponential, trigonometric and logarithmic functions, or to inhomogeneous polynomials, must be dimensionless quantities. (Note: this requirement is somewhat relaxed in Siano's orientational analysis described below, in which the square of certain dimensioned quantities are dimensionless) This requirement is clear when one observes the Taylor expansions for these functions (a sum of various powers of the function argument). For example, the logarithm of 3 kg is undefined even though the logarithm of 3 is nearly 0.477. An attempt to compute ln 3 kg would produce, if one naively took ln 3 kg to mean the dimensionally meaningless "ln (1 + 2 kg)", which is dimensionally incompatible the sum has no meaningful dimension requiring the argument of transcendental functions to be dimensionless.

Another way to understand this problem is that the different coefficients scale differently under change of units were one to reconsider this in grams as "ln 3000 g" instead of "ln 3 kg", one could compute ln 3000, but in terms of the Taylor series, the degree 1 term would scale by 1000, the degree-2 term would scale by 10002, and so forth the overall output would not scale as a particular dimension. While most mathematical identities about dimensionless numbers translate in a straightforward manner to dimensional quantities, care must be taken with logarithms of ratios: the identity log(a/b) = log a log b, where the logarithm is taken in any base, holds for dimensionless numbers a and b, but it does not hold if a and b are dimensional, because in this case the left-hand side is well-defined but the right-hand side is not. Similarly, while one can evaluate monomials (xn) of dimensional quantities, one cannot evaluate polynomials of mixed degree with dimensionless coefficients on dimensional quantities: for x2, the expression (3 m)2 = 9 m2 makes sense (as an area), while for x2 + x, the expression (3 m)2 + 3 m = 9 m2 + 3 m does not make sense. However, polynomials of mixed degree can make sense if the coefficients are suitably chosen physical quantities that are not dimensionless. For example, This is the height to which an object rises in time t if the acceleration of gravity is 32 feet per second per second and the initial upward speed is 500 feet per second. It is not even necessary for t to be in seconds. For example, suppose t = 0.01 minutes. Then the first term would be 15 Dimensional analysis 16 Incorporating units The value of a dimensional physical quantity Z is written as the product of a unit [Z] within the dimension and a dimensionless numerical factor, n. In a strict sense, when like-dimensioned quantities are added or subtracted or compared, these dimensioned quantities must be expressed in consistent units so that the numerical values of these quantities may be directly added or subtracted. But, in concept, there is no problem adding quantities of the same dimension expressed in different units. For example, 1 meter added to 1 foot is a length, but it would not be correct to add 1 to 1 to get the result. A conversion factor, which is a ratio of like-dimensioned quantities and is equal to the dimensionless unity, is needed: is identical to The factor is identical to the dimensionless 1, so multiplying by this conversion factor changes nothing. Then when adding two quantities of like dimension, but expressed in different units, the appropriate conversion factor, which is essentially the dimensionless 1, is used to convert the quantities to identical units so that their

numerical values can be added or subtracted. Only in this manner is it meaningful to speak of adding like-dimensioned quantities of differing units. Position vs displacement Some discussions of dimensional analysis implicitly describes all quantities are mathematical vectors. (In mathematics scalars are considered a special case of vectors; the emphasis here is that vectors are closed under addition, subtraction, and scalar multiplication, and permit scalar division.). This assumes an implicit point of referencean origin. While this is useful and often perfectly adequate, allowing many important errors to be caught, it can fail to model certain aspects of physics. A more rigorous approach requires distinguishing between position and displacement (or moment in time versus duration, or absolute temperature versus temperature change). Consider points on a line, each with a position with respect to a given origin, and distances among them. Positions and displacements all have units of length, but their meaning is not interchangeable: adding two displacements should yield a new displacement (walking ten paces then twenty paces gets you thirty paces forward), adding a displacement to a position should yield a new position (walking one block down the street from an intersection gets you to the next intersection), subtracting two positions should yield a displacement, but one may not add two positions. This illustrates the subtle distinction between affine quantities (ones modeled by an affine space, such as position) and vector quantities (ones modeled by a vector space, such as displacement). Vector quantities may be added to each other, yielding a new vector quantity, and a vector quantity may be added to a suitable affine quantity (a vector space acts on an affine space), yielding a new affine quantity. Affine quantities cannot be added, but may be subtracted, yielding relative quantities which are vectors, and these relative differences may then be added to each other or to an affine quantity. Dimensional analysis 17 Properly then, positions have dimension of affine length, while displacements have dimension of vector length. To assign a number to an affine unit, one must not only choose a unit of measurement, but also a point of reference, while to assign a number to a vector unit only requires a unit of measurement. Thus some physical quantities are better modeled by vectorial quantities while others tend to require affine representation, and the distinction is reflected in their dimensional analysis. This distinction is particularly important in the case of temperature for which there is an absolute zero that is different in different measuring systems. That is, for absolute temperatures 0 K = 273.15 C = 459.67 F = 0 R, but for relative temperatures, 1 K = 1 C 1 F = 1 R

Unit conversion for relative temperatures, where no temperature difference is zero in all units, is simply a matter of multiplying by, e.g., 1 F / 1 K. But because these systems for absolute temperatures have different origins, conversion from one absolute temperature requires accounting for that. As a result, simple dimensional analysis can still lead to errors if it becomes ambiguous if 1 K equals 274.15 C or 1 C. Orientation and frame of reference Similar to the issue of a point of reference is the issue of orientation: a displacement in 2 or 3 dimensions is not just a length, but is a length together with a direction. (This issue does not arise in 1 dimension, or rather is equivalent to the distinction between positive and negative.) Thus, to compare or combine two dimensional quantities in a multi-dimensional space, one also needs an orientation: they need to be compared to a frame of reference. This leads to the extensions discussed below, namely Huntley's directed dimensions and Siano's orientational analysis. Other uses Dimensional analysis is also used to derive relationships between the physical quantities that are involved in a particular phenomenon that one wishes to understand and characterize. It was used for the first time (Pesic, 2005) in this way in 1872 by Lord Rayleigh, who was trying to understand why the sky is blue. Examples A simple example: period of a harmonic oscillator What is the period of oscillation of a mass attached to an ideal linear spring with spring constant suspended in gravity of strength ? The four quantities have the following dimensions: [T]; [M]; ; and variables, . From these we can form only one dimensionless product of powers of our chosen = . The dimensionless product of powers of variables is sometimes referred to as a dimensionless group of variables, but the group, , referred to means "collection" rather than mathematical group. They are often called dimensionless numbers as well. Note that no other dimensionless product of powers involving with k, m, T, and g alone can be formed, because only g involves L . Dimensional analysis can sometimes yield strong statements about the irrelevance of some quantities in a problem, or the need for additional parameters. If we have chosen enough variables to properly describe the problem, then from this argument we can conclude that the period of the mass on the spring is independent of g: it is the same on the earth or the moon. The equation demonstrating the existence of a product of powers for our problem can be written in an entirely equivalent way: , for some dimensionless constant .

Dimensional analysis When faced with a case where our analysis rejects a variable (g, here) that we feel sure really belongs in a physical description of the situation, we might also consider the possibility that the rejected variable is in fact relevant, and that some other relevant variable has been omitted, which might combine with the rejected variable to form a dimensionless quantity. That is, however, not the case here. When dimensional analysis yields a solution of problems where only one dimensionless product of powers is involved, as here, there are no unknown functions, and the solution is said to be "complete." A more complex example: energy of a vibrating wire Consider the case of a vibrating wire of length l (L) vibrating with an amplitude A (L). The wire has a linear density (M/L) and is under tension s (ML/T2), and we want to know the energy E (ML2/T2) in the wire. Let 1 and 2 be two dimensionless products of powers of the variables chosen, given by The linear density of the wire is not involved. The two groups found can be combined into an equivalent form as an equation where F is some unknown function, or, equivalently as where f is some other unknown function. Here the unknown function implies that our solution is now incomplete, but dimensional analysis has given us something that may not have been obvious: the energy is proportional to the first power of the tension. Barring further analytical analysis, we might proceed to experiments to discover the form for the unknown function f. But our experiments are simpler than in the absence of dimensional analysis. We'd perform none to verify that the energy is proportional to the tension. Or perhaps we might guess that the energy is proportional to l, and so infer that E = ls. The power of dimensional analysis as an aid to experiment and forming hypotheses becomes evident. The power of dimensional analysis really becomes apparent when it is applied to situations, unlike those given above, that are more complicated, the set of variables involved are not apparent, and the underlying equations hopelessly complex. Consider, for example, a small pebble sitting on the bed of a river. If the river flows fast enough, it will actually raise the pebble and cause it to flow along with the water. At what critical velocity will this occur? Sorting out the guessed variables is not so easy as before. But dimensional analysis can be a powerful aid in understanding problems like this, and is usually the very first tool to be applied to complex problems where the underlying equations and constraints are poorly understood. In such cases, the answer may depend on a dimensionless number such as the Reynolds number, which may be interpreted by dimensional analysis. Extensions Huntley's extension: directed dimensions

Huntley (Huntley, 1967) has pointed out that it is sometimes productive to refine our concept of dimension. Two possible refinements are: The magnitude of the components of a vector are to be considered dimensionally distinct. For example, rather than an undifferentiated length unit L, we may have represent length in the x direction, and so forth. This requirement stems ultimately from the requirement that each component of a physically meaningful equation (scalar, vector, or tensor) must be dimensionally consistent. Mass as a measure of quantity is to be considered dimensionally distinct from mass as a measure of inertia. 18 Dimensional analysis 19 As an example of the usefulness of the first refinement, suppose we wish to calculate the distance a cannon ball travels when fired with a vertical velocity component and a horizontal velocity component , assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are then dimensioned as , , both , R, the distance travelled, having dimension L, and g the downward acceleration of gravity, with dimension With these four quantities, we may conclude that the equation for the range R may be written: Or dimensionally from which we may deduce that and , which leaves one exponent undetermined. This is to be expected since we have two fundamental quantities L and T and four parameters, with one equation. If, however, we use directed length dimensions, then and g as will be dimensioned as , as , R as . The dimensional equation becomes: and we may solve completely as , and . The increase in deductive power gained by the use of directed length dimensions is apparent. In a similar manner, it is sometimes found useful (e.g., in fluid mechanics and thermodynamics) to distinguish between mass as a measure of inertia (inertial mass), and mass as a measure of quantity (substantial mass). For example, consider the derivation of Poiseuille's Law. We wish to find the rate of mass flow of a viscous fluid through a circular pipe. Without drawing distinctions between inertial and substantial mass we may

choose as the relevant variables the mass flow rate with dimensions the pressure gradient along the pipe with dimensions the density with dimensions the dynamic fluid viscosity with dimensions the radius of the pipe with dimensions There are three fundamental variables so the above five equations will yield two dimensionless variables which we may take to be and and we may express the dimensional equation as where C and a are undetermined constants. If we draw a distinction between inertial mass with dimensions substantial mass with dimensions and , then mass flow rate and density will use substantial mass as the mass parameter, while the pressure gradient and coefficient of viscosity will use inertial mass. We now have four fundamental parameters, and one dimensionless constant, so that the dimensional equation may be written: where now only C is an undetermined constant (found to be equal to by methods outside of dimensional analysis). This equation may be solved for the mass flow rate to yield Poiseuille's law. Dimensional analysis Siano's extension: orientational analysis Huntley's extension has some serious drawbacks: It does not deal well with vector equations involving the cross product, nor does it handle well the use of angles as physical variables. It also is often quite difficult to assign the L, Lx, Ly, Lz, symbols to the physical variables involved in the problem of interest. He invokes a procedure that involves the "symmetry" of the physical problem. This is often very difficult to apply reliably: It is unclear as to what parts of the problem that the notion of "symmetry" is being invoked. Is it the symmetry of the physical body that forces are acting upon, or to the points, lines or areas at which forces are being applied? What if more than one body is involved with different symmetries? Consider the spherical bubble attached to a cylindrical tube, where one wants the flow rate of air as a function of the pressure difference in the two parts. What are the Huntley extended dimensions of the viscosity of the air contained in the connected parts? What are the extended dimensions of the pressure of the two parts? Are they the same or different? These difficulties are responsible for the limited application of Huntley's addition to real problems. Angles are, by convention, considered to be dimensionless variables, and so the use of angles as

physical variables in dimensional analysis can give less meaningful results. As an example, consider the projectile problem mentioned above. Suppose that, instead of the x- and y-components of the initial velocity, we had chosen the magnitude of the velocity v and the angle at which the projectile was fired. The angle is, by convention, considered to be dimensionless, and the magnitude of a vector has no directional quality, so that no dimensionless variable can be composed of the four variables g, v, R, and . Conventional analysis will correctly give the powers of g and v, but will give no information concerning the dimensionless angle . Siano (Siano, 1985-I, 1985-II) has suggested that the directed dimensions of Huntley be replaced by using orientational symbols 1x 1y 1z to denote vector directions, and an orientationless symbol 10. Thus, Huntley's 1x becomes L 1x with L specifying the dimension of length, and 1x specifying the orientation. Siano further shows that the orientational symbols have an algebra of their own. Along with the requirement that 1i1 = 1i, the following multiplication table for the orientation symbols results: Note that the orientational symbols form a group (the Klein four-group or "Viergruppe"). In this system, scalars always have the same orientation as the identity element, independent of the "symmetry of the problem." Physical quantities that are vectors have the orientation expected: a force or a velocity in the z-direction has the orientation of 1z. For angles, consider an angle that lies in the z-plane. Form a right triangle in the z plane with being one of the acute angles. The side of the right triangle adjacent to the angle then has an orientation 1x and the side opposite has an orientation 1y. Then, since tan() = 1y/1x = + ... we conclude that an angle in the xy plane must have an orientation 1y/1x = 1z, which is not unreasonable. Analogous reasoning forces the conclusion that sin() has orientation 1z while cos() has orientation 10. These are different, so one concludes (correctly), for example, that there are no solutions of physical equations that are of the form a cos()+b sin() , where a and b are real scalars. Physical quantities may be expressed as complex numbers (e.g. ) which imply that the complex quantity i has an orientation equal to that of the angle it is associated with (1z in the above example). The assignment of orientational symbols to physical quantities and the requirement that physical equations be orientationally homogeneous can actually be used in a way that is similar to dimensional analysis to derive a little more information about acceptable solutions of physical problems. In this approach one sets up the dimensional equation and solves it as far as one can. If the lowest power of a physical variable is fractional, both sides of the solution is raised to a power such that all powers are integral. This puts it into "normal form". The orientational

20 Dimensional analysis equation is then solved to give a more restrictive condition on the unknown powers of the orientational symbols, arriving at a solution that is more complete than the one that dimensional analysis alone gives. Often the added information is that one of the powers of a certain variable is even or odd. As an example, for the projectile problem, using orientational symbols, , being in the xy-plane will thus have dimension 1z and the range of the projectile R will be of the form: Dimensional homogeneity will now correctly yield a = 1 and b = 2, and orientational homogeneity requires that c be an odd integer. In fact the required function of theta will be sin()cos() which is a series of odd powers of . It is seen that the Taylor series of sin() and cos() are orientationally homogeneous using the above multiplication table, while expressions like cos() + sin() and exp() are not, and are (correctly) deemed unphysical. It should be clear that the multiplication rule used for the orientational symbols is not the same as that for the cross product of two vectors. The cross product of two identical vectors is zero, while the product of two identical orientational symbols is the identity element. Percentages and derivatives Percentages are dimensionless quantities, since they are ratios of two quantities with the same dimensions. Derivatives with respect to a quantity add the dimensions of the variable one is differentiating with respect to on the denominator. Thus: position (x) has units of L (Length); derivative of position with respect to time (dx/dt, velocity) has units of L/T Length from position, Time from the derivative; the second derivative (d2x/dt2, acceleration) has units of L/T2. In economics, one distinguishes between stocks and flows: a stock has units of "units" (say, widgets or dollars), while a flow is a derivative of a stock, and has units of "units/time" (say, dollars/year). Beware that in some contexts, dimensional quantities are expressed as dimensionless quantities or percentages by omitting some dimensions. This may or may not be misleading. For example, Debt to GDP ratios are generally expressed as percentages: total debt outstanding (dimension of Currency) divided by annual GDP (dimension of Currency) but one may argue that in comparing a stock to a flow, annual GDP should have dimensions of Currency/Time (Dollars/Year, for instance), and thus Debt to GDP should have units of years. Dimensionless concepts Constants The dimensionless constants that arise in the results obtained, such as the C in the Poiseuille's Law problem and the in the spring problems discussed above come from a more detailed analysis of the underlying physics, and often

arises from integrating some differential equation. Dimensional analysis itself has little to say about these constants, but it is useful to know that they very often have a magnitude of order unity. This observation can allow one to sometimes make "back of the envelope" calculations about the phenomenon of interest, and therefore be able to more efficiently design experiments to measure it, or to judge whether it is important, etc. 21 Dimensional analysis 22 Formalisms Paradoxically, dimensional analysis can be a useful tool even if all the parameters in the underlying theory are dimensionless, e.g., lattice models such as the Ising model can be used to study phase transitions and critical phenomena. Such models can be formulated in a purely dimensionless way. As we approach the critical point closer and closer, the distance over which the variables in the lattice model are correlated (the so-called correlation length, ) becomes larger and larger. Now, the correlation length is the relevant length scale related to critical phenomena, so one can, e.g., surmize on "dimensional grounds" that the non-analytical part of the free energy per lattice site should be where is the dimension of the lattice. It has been argued by some physicists, e.g., Michael Duff,[4] [5] that the laws of physics are inherently dimensionless. The fact that we have assigned incompatible dimensions to Length, Time and Mass is, according to this point of view, just a matter of convention, borne out of the fact that before the advent of modern physics, there was no way to relate mass, length, and time to each other. The three independent dimensionful constants: c, , and G, in the fundamental equations of physics must then be seen as mere conversion factors to convert Mass, Time and Length into each other. Just as in the case of critical properties of lattice models, one can recover the results of dimensional analysis in the appropriate scaling limit; e.g., dimensional analysis in mechanics can be derived by reinserting the constants , c, and G (but we can now consider them to be dimensionless) and demanding that a nonsingular relation between quantities exists in the limit , and . In problems involving a gravitational field the latter limit should be taken such that the field stays finite. Applications Dimensional analysis is most often used in physics and chemistry- and in the mathematics thereofbut finds some applications outside of those fields as well. Mathematics

A simple application of dimensional analysis to mathematics is in computing the form of the volume of an n-ball (the solid ball in n-dimensions), or the area of its surface, the n-sphere: being an n-dimensional figure, the volume scales as while the surface area, being -dimensional, scales as Thus the volume of the n-ball in terms of the radius is for some constant Determining the constant takes more involved mathematics, but the form can be deduced and checked by dimensional analysis alone. Proof of the Pythagorean theorem A very simple proof of the Pythagorean theorem can be obtained by just dimensional reasoning.[6] The area of any triangle depends on its size and shape, which can be unambiguously identified by the length of one of its edges (for example, the largest) and by any two of its angles (the third being determined by the fact that the sum of all three is ). Thus, recalling that an area has the dimensions of a length squared, we can write: Dimensional proof of the Pythagorean theorem Dimensional analysis area = largest_edge2 f (angle_1, angle_2), where f is an adimensional function of the angles. Now, referring to the figure at right, if we divide a right triangle in two smaller ones by tracing the segment perpendicular to its hypotenuse and passing by the opposite vertex, and express the obvious fact that the total area is the sum of the two smaller ones, by applying the previous equation we have: c2 f (, /2) = a2 f (, /2) + b2 f (, /2). And, eliminating f: c2 = a2 + b2 , Q.E.D. Note that the result is obtained without specifying the form of the adimensional function f. Finance, economics, and accounting In finance, economics, and accounting, dimensional analysis is most commonly used in interpreting various financial ratios, economics ratios, and accounting ratios. For example, the P/E ratio has dimensions of time (units of years), and can be interpreted as "years of earnings to earn the price paid." In economics, debt-to-GDP ratio also has units of years (debt has units of currency, GDP has units of currency/year). More surprisingly, bond duration also has units of years, which can be shown by dimensional analysis, but takes some financial intuition to understand.

Velocity of money has units[7] of 1/Years (GDP/Money supply has units of Currency/Year over Currency): how often a unit of currency circulates per year. Dimensional analysis is rarely used in (mainstream/neoclassical) economic modeling,[8] and economic models are often dimensionally inconsistent.[9] The equation of exchange is the most notable example of a dimensional equation in economic modeling,[8] while the widely-used CobbDouglas model does not use dimensions in a meaningful way.[10] This lack of dimensional consistency is criticized by heterodox economics, notably Austrian economics,[11] while dimensional consistency is not considered necessary or desirable by mainstream economists. [9] [12] See also Quantity calculus Debt to GDP ratio Denominate number Dirac large numbers hypothesis Fermi problem Fundamental unit Nondimensionalization Equivalization Physical quantity Natural units Similitude (model) Buckingham theorem Units conversion by factor-label Affine space Vector space Frame of reference Point of reference 23 Dimensional analysis Rayleigh's method of dimensional analysis

Covariance and contravariance of vectors Wedge product Geometric algebra Notes [1] Stahl, Walter R (1961), "Dimensional Analysis In Mathematical Biology", Bulletin of Mathematical Biophysics 23: 355 [2] Roche, John J (1998), The Mathematics of Measurement: A Critical History, London: Springer, p. 203, ISBN 978-0387915814, " Beginning apparently with Maxwell, mass, length and time began to be interpreted as having a privileged fundamental character and all other quantities as derivative, not merely with respect to measurement, but with respect to their physical status as well. (http:/ / books. google. com/ books?id=eiQOqS-Q6EkC& lpg=PP1& dq=isbn:978-0387915814& pg=PA203#v=onepage& q& f=false)" [3] Mason, Stephen Finney (1962), A history of the sciences, New York: Collier Books, p. 169, ISBN 0-02-093400-9 [4] M. J. Duff, L. B. Okun and G. Veneziano, Trialogue on the number of fundamental constants, JHEP 0203, 023 (2002) preprint (http:/ / arxiv. org/ abs/ physics/ 0110060). [5] M. J. Duff,Comment on time-variation of fundamental constants, preprint (http:/ / arxiv. org/ abs/ hep-th/ 0208093) [6] see, for example: F.Olness (1997), Dimensional Regularization Meets Freshman E&M (http:/ / www. physics. smu. edu/ ~olness/ cteqpp/ DimReg. 5. pdf) [7] "It's just a flesh wound..." (http:/ / www. debtdeflation. com/ blogs/ 2009/ 03/ 17/ its-just-aflesh-wound/ ), Steve Keen [8] (Barnett 2007, footnote 8, p. 96) [9] "And, from referee #3s report: 'There is no question that the lack of dimensional consistency is pervasive throughout mathematical economics. However, this paper does not make clear why this lack of dimensional consistency is problematical. The lack of dimensional consistency is not so much a problem in and of itself . . .'", (Barnett 2007, p. 101, referee report #3) [10] (Barnett 2007, p. 96) [11] (Barnett 2007) [12] Four mainstream economists at a leading journal are quoted in (Barnett 2007, Appendix, pp. 99102) as stating that dimensional consistency is not necessary in economic modeling and lack of dimensional consistency is not a valid criticism of an economic model. References Barenblatt, G. I. (1996), Scaling, Self-Similarity, and Intermediate Asymptotics, Cambridge, UK: Cambridge University Press, ISBN 0-521-43522-6 Barnett (2007), "Dimensions and Economics: Some Problems" (http://mises.org/journals/qjae/pdf/ qjae7_1_10.pdf), Quarterly Journal of Austrian Economics 7 (1) Bhaskar, R.; Nigam, Anil (1990), "Qualitative Physics Using Dimensional Analysis", Artificial Intelligence 45: 73111, doi:10.1016/0004-3702(90)90038-2 Bhaskar, R.; Nigam, Anil (1991), "Qualitative Explanations of Red Giant Formation", The Astrophysical Journal 372: 5926, doi:10.1086/170003 Boucher; Alves (1960), "Dimensionless Numbers", Chem. Eng. Progress 55: 5564 Bridgman, P. W. (1922), Dimensional Analysis, Yale University Press, ISBN 0548910294

Buckingham, Edgar (1914), "On Physically Similar Systems: Illustrations of the Use of Dimensional Analysis", Phys. Rev. 4: 345, doi:10.1103/PhysRev.4.345 Hart, George W. (March 1 1995), Multidimensional Analysis: Algebras and Systems for Science and Engineering (http://www.georgehart.com/research/multanal.html), Springer-Verlag, ISBN 0-387-94417-6 Huntley, H. E. (1967), Dimensional Analysis, Dover, LOC 67-17978 Klinkenberg, A. (1955), " ", Chem. Eng. Science 4: 130140, 167177, doi:10.1016/00092509(55)80004-8 Langhaar, H. L. (1951), Dimensional Analysis and Theory of Models, Wiley, ISBN 0882756826 Moody, L. F. (1944), "Friction Factors for Pipe Flow", Trans. Am. Soc. Mech. Engrs. 66 (671) Murphy, N. F. (1949), "Dimensional Analysis", Bull. V.P.I. 42 (6) Perry, J. H.; et al. (1944), "Standard System of Nomenclature for Chemical Engineering Unit Operations", Trans. Am. Inst. Chem. Engrs. 40 (251) Pesic, Peter (2005), Sky in a Bottle, Cambridge, Mass: MIT Press, pp. 2278, ISBN 0-262-162342 24 Dimensional analysis Petty, G. W. (2001), "Automated computation and consistency checking of physical dimensions and units in scientific programs.", Software Practice and Experience 31: 106776, doi:10.1002/spe.401 Porter, Alfred W. (1933), The Method of Dimensions, Methuen Lord Rayleigh (1915), "The Principle of Similitude", Nature 95: 668, doi:10.1038/095066c0 Siano, Donald (1985), "Orientational Analysis A Supplement to Dimensional Analysis I", J. Franklin Institute 320 (320): 267, doi:10.1016/0016-0032(85)90031-6 Siano, Donald (1985), "Orientational Analysis, Tensor Analysis and The Group Properties of the SI Supplementary Units II", J. Franklin Institute 320 (320): 285, doi:10.1016/0016-0032(85)900328 Silberberg, I. H.; McKetta J. J. Jr. (1953), "Learning How to Use Dimensional Analysis", Petrol. Refiner 32 (4 (p.5), 5(p.147), 6(p.101), 7(p.129)) Van Driest, E. R. (March 1946), "On Dimensional Analysis and the Presentation of Data in Fluid Flow Problems", J. App. Mech 68 (A-34) Whitney, H. (1968), "The Mathematics of Physical Quantities, Parts I and II" (http://jstor.org/stable/2315883), Am. Math. Mo. (Mathematical Association of America) 75 (2): 115138, 227256, doi:10.2307/2315883 GA Vignaux (1992), Erickson, Gary J.; Neudorfer, Paul O., ed., Dimensional Analysis in Data Modelling, Kluwer Academic, ISBN 0-7923-2031-X Wacaw Kasprzak, Bertold Lysik, Marek Rybaczuk (1990), Dimensional Analysis in the Identification of Mathematical Models, World Scientific, ISBN 9789810203047 PF Mendez, F Ordez (September 2005), "Scaling Laws From Statistical Data and Dimensional Analysis", Journal of Applied Mechanics 72 (5): 648657, doi:10.1115/1.1943434 G Hart (1994), The theory of dimensioned matrices S. Drobo (1954), "On the foundations of dimensional analysis", Studia Mathematica

External links List of dimensions for variety of physical quantities (http://www.roymech.co.uk/Related/Fluids/ Dimension_Analysis.html) Unicalc Live web calculator doing units conversion by dimensional analysis (http://www.calchemy.com/ uclive.htm) http://www.math.ntnu.no/~hanche/notes/buckingham/buckingham-a4.pdf http://rain.aos.wisc.edu/~gpetty/physunits.html Quantity System calculator for units conversion based on dimensional approach (http://QuantitySystem. CodePlex.com) Units, quantities, and fundamental constants project dimensional analysis maps (http://www. outlawmapofphysics.com) 25 Physics 26 Physics Physics (from Ancient Greek: physis "nature") is a natural science that involves the study of matter[1] and its motion through spacetime, as well as all related concepts, including energy and force.[2] More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.[3] [4] [5] Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy.[6] Over the last two millenia, physics was a part of natural philosophy along with chemistry, certain branches of mathematics, and biology, but during the Scientific Revolution in the 16th century, the natural sciences emerged as unique research programs in their own right.[7] Certain research areas are interdisciplinary, such as mathematical physics and quantum chemistry, which means that the boundaries of physics are not rigidly defined. In the nineteenth and twentieth centuries physicalism emerged as a major unifying feature of the philosophy of science as physics provides fundamental explanations for every observed natural phenomenon. New ideas in physics often explain the fundamental mechanisms of other sciences, while opening to new research areas in mathematics and philosophy. Physics is also significant and influential through advances in its understanding that have translated into new technologies. For example, advances in the understanding of electromagnetism or nuclear physics led directly to the development of new products which have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus. Scope and aims Physics covers a wide range of phenomena, from elementary particles (such as quarks, neutrinos and electrons) to the largest superclusters of galaxies. Included in these phenomena are the most basic objects from which all other things are composed, and therefore physics is

sometimes called the "fundamental science".[8] Physics aims to describe the various phenomenon that occur in nature in terms of simpler phenomena. Thus, physics aims to both connect the things observable to humans to root causes, and then to try to connect these causes together. For example, the ancient Chinese observed that certain rocks (lodestone) were attracted to one another by some invisible force. This effect was later called magnetism, and was first rigorously studied in the 17th century. A little earlier than the Chinese, the ancient Greeks knew of other objects such as amber, that when rubbed with fur would cause a similar invisible attraction between the two. This was also first studied rigorously in the 17th century, and came to be called electricity. Thus, physics had come to understand two observations of nature in terms of some root cause (electricity and magnetism). However, further work in the 19th century revealed that these two forces were just two different aspects of one force electromagnetism. This process of "unifying" forces continues today, and electromagnetism and the weak nuclear force are now considered to be two aspects of the electroweak interaction. Physics hopes to find an ultimate reason (Theory of Everything) for why nature is as it is (see section Current research below for more information). Physics involves modeling the natural world with abstract theory. Here, the path of a particle is modeled with the mathematics of calculus to explain its behavior: the purview of the branch of physics known as mechanics. Physics 27 Scientific method Physicists use a scientific method to test the validity of a physical theory, using a methodical approach to compare the implications of the theory in question with the associated conclusions drawn from experiments and observations conducted to test it. Experiments and observations are to be collected and matched with the predictions and hypotheses made by a theory, thus aiding in the determination or the validity/invalidity of the theory. Theories which are very well supported by data and have never failed any competent empirical test are often called scientific laws, or natural laws. Of course, all theories, including those called scientific laws, can always be replaced by more accurate, generalized statements if a disagreement of theory with observed data is ever found.[9] Theory and experiment Theorists seek to develop mathematical models that both agree with existing experiments and successfully predict future results, while experimentalists devise and perform experiments to test theoretical predictions and explore new phenomena. Although theory and experiment are developed separately, they are strongly dependent upon each other. Progress in physics frequently comes about when experimentalists make a discovery that existing theories cannot

explain, or when new theories generate experimentally testable predictions, which inspire new experiments. Physicists who work at the interplay of theory and experiment are called phenomenologists. Phenomenologists look at the complex phenomena observed in experiment and work to relate them to fundamental theory. The astronaut and Earth are both in free-fall Theoretical physics has historically taken inspiration from philosophy; electromagnetism was unified this way.[10] Beyond the known universe, the field of theoretical physics also deals with hypothetical issues,[11] such as parallel universes, a multiverse, and higher dimensions. Theorists invoke these ideas in hopes of solving particular problems with existing theories. They then explore the consequences of these ideas and work toward making testable predictions. Experimental physics informs, and is informed by, engineering and technology. Experimental physicists involved in basic research design and perform experiments with equipment such as particle accelerators Lightning is an electric current and lasers, whereas those involved in applied research often work in industry, developing technologies such as magnetic resonance imaging (MRI) and transistors. Feynman has noted that experimentalists may seek areas which are not well explored by theorists.[12] Physics 28 Relation to mathematics and the other sciences In the Assayer (1622), Galileo noted that mathematics is the language in which Nature expresses its laws.[13] Most experimental results in physics are numerical measurements, and theories in physics use mathematics to give numerical results to match these measurements. This parabola-shaped lava flow illustrates Galileo's law of falling bodies as well as blackbody radiation the temperature is discernible from the color of the blackbody. Physics relies upon mathematics to provide the logical framework in which physical laws may be precisely formulated and predictions quantified. Whenever analytic solutions of equations are not feasible, numerical analysis and simulations may be utilized. Thus, scientific computation is an integral part of physics, and the field of computational physics is an active area of research. A key difference between physics and mathematics is that since physics is ultimately concerned with descriptions of the material world, it tests it