:: wikimiki.org ::
| Dimensional Analysis |
Dimensional analysisDimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. It is routinely used by physical scientists and engineers to check the plausibility of derived equations. Only like dimensioned quantities may be added, subtracted, compared, or equated. When unlike dimensioned quantities appear opposite of the "+" or "−" or "=" sign, that physical equation is not plausible, which might prompt one to correct errors before proceeding to use it. When like dimensioned quantities or unlike dimensioned quantities are multiplied or divided, their dimensions are likewise multiplied or divided. When dimensioned quantities are raised to a power or a power root, the same is done to the dimensions attached to those quantities.
__TOC__
The dimensions of a physical quantity is associated with symbols, such as M, L, T which represent mass, length and time, each raised to rational powers. Just as there are derived units that are derived from base units, there are primary base dimensions of physical quantity and secondary dimensions of physical quantity derived from the former. For instance, the dimension of the physical quantity, speed, is distance/time (L/T) and the dimension of a force is mass × distance/time² or ML/T². In mechanics, every dimension of physical quantity can be expressed in terms of distance (which physicists often call "length"), time, and mass, or alternatively in terms of force, length and mass. Depending on the problem, it may be advantageous to choose one or another other set of base dimensions. In electromagnetism, for example, it may be useful to use dimensions of M, L, T, and Q, where Q represents quantity of electric charge.
The units of a physical quantity are defined by convention, related to some standard; e.g. length may have units of meters, feet, inches, miles or micrometres; but a length always has a dimension of L whether it is measured in meters, feet, inches, miles or micrometres. Dimensional symbols, such as L, form a group: there is an identity, 1; there is an inverse to L, which is 1/L, and L raised to any rational power p is a member of the group, having an inverse of 1/L raised to the power p. There are conversion factors between units; for example one meter is equal to 39.37 inches, but a meter and an inch are both associated with the same symbol, L.
In the most primitive form, dimensional analysis may be used to check the correctness of physical equations: in every physically meaningful expression, only quantities of the same dimension can be added or subtracted. Moreover, the two sides of any equation must have the same dimensions. For example, the mass of a rat and the mass of a flea may be added, but the mass of a flea and the length of a rat cannot be added. Furthermore, the arguments to exponential, trigonometric and logarithmic functions must be dimensionless numbers. The logarithm of 3 kg is undefined, but the logarithm of 3 is nearly 0.477.
It should be noted that sometimes different physical quantities may have the same dimensions: work (or energy) and torque, for example, both have the same dimensions, ML²/T². An equation with torque on one side and energy on the other would be dimensionally correct, but cannot be physically correct! However, torque multiplied by an angular twist measured in (dimensionless) radians is work or energy. (The radian is the mathematically natural measure of an angle and is the ratio of arc of a circle swept by such an angle divided by the radius of the circle. That ratio of like dimensioned quantities, length over length, is dimensionless.)
The value of a dimensionful physical quantity is written and thought of as the product of a unit within the dimension and a dimensionless numerical factor. Strictly, 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, conceptually, 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 saying
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, it is meaningful to speak of adding like dimensioned quantities of differing units, although to do so mathematically, all units must be the same. It is not meaningful, either physically or mathematically, to speak of adding unlike dimensioned physical quantities such as adding length (say, in meters) to mass (perhaps in kilograms).
Buckingham π theorem
The Buckingham π theorem forms the basis of the central tool of dimensional analysis. 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, even if the form of the equation is still unknown.
A worked example
A typical application of dimensional analysis occurs in fluid dynamics. If a moving fluid meets an object, it exerts a force on the object, according to a complicated (and not completely understood) law. We might suppose that the variables involved under some conditions to be the speed, density and viscosity of the fluid, the size of the body (expressed in terms of its frontal area ), and the drag force. Using the algorithm of the π-theorem, one can reduce these five variables to two dimensionless parameters: the drag coefficient and the Reynolds number.
Alternatively, one can derive the dimensionless parameters via direct
manipulation of the underlying differential equations.
That this is so becomes obvious when the drag force is expressed as part of a function of the other variables in the problem:
:
This rather odd form of expression is used because it does not assume a one-one relationship. Here, is some function (as yet unknown) that takes five arguments. We note that the right hand side is zero in any system of units; so it should be possible to express the relationship described by in terms of only dimensionless groups.
There are many ways of combining the five arguments of to form dimensionless groups, but the Buckingham Pi theorem states that there will be two such groups. The most appropriate are the Reynolds number, given by
:
and the drag coefficient, given by
:
Thus the original law involving a function of five variables may be replaced by one involving only two:
:
where is some function of two arguments.
The original law is then reduced to a law involving only these two numbers.
Because the only unknown in the above equation is , it is possible to express it as
:
or
:
Thus the force is simply times some (as yet unknown) function of the Reynolds number: a considerably simpler system than the original five-argument function given above.
Dimensional analysis thus makes a very complex problem (trying to determine the behavior of a function of five variables) a much simpler one: the determination of the drag as a function of only one variable, the Reynolds number.
The analysis also gives other information for free, so to speak. We know that, other things being equal, the drag force will be proportional to the density of the fluid. This kind of information often proves to be extremely valuable, especially in the early stages of a research project.
To empirically determine the Reynolds number dependence, instead of experimenting on huge bodies with fast flowing fluids (such as real-size airplanes in wind-tunnels), one may just as well experiment on small models with slow flowing, more viscous fluids, because these two systems are similar.
Another simple example
Dimensional analysis can sometimes yield strong statements about the irrelevance of some quantities in a problem.
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]; [M/T^2]; and [L/T^2]. From these we can form only one dimensionless group, .
There is no other group involving . Thus the period of the mass on the spring is the same on the earth or the moon. Indeed, dimensional analysis tells us that , for some dimensionless constant .
See also
- Fundamental unit
- Similitude (model)
- Buckingham Pi theorem
- Dimensionless number
References
- Barenblatt, G. I., "Scaling, Self-Similarity, and Intermediate Asymptotics", Cambridge University Press, 1996
- Bridgman, P. W., "Dimensional Analysis", Yale University Press, 1937
- Langhaar, H. L., "Dimensional Analysis and Theory of Models", Wiley, 1951
- Murphy, N. F., Dimensional Analysis, Bull. V.P.I., 1949, 42(6)
- Porter, "The Method of Dimensions", Methuen, 1933
- Boucher and Alves, Dimensionless Numbers, Chem. Eng. Progress, 1960, 55, pp.55-64
- Buckingham, E., On Physically Similar Systems: Illustrations of the Use of Dimensional Analysis, Phys. Rev, 1914, 4, p.345
- Klinkenberg A. Chem. Eng. Science, 1955, 4, pp. 130-140, 167-177
- Rayleigh, Lord, The Principle of Similitude, Nature 1915, 95, pp. 66-68
- Silberberg, I. H. and McKetta J. J., Jr., Learning How to Use Dimensional Analysis, Petrol. Refiner, 1953, 32(4), p179; (5), p.147; (6), p.101; (7), p. 129
- Van Driest, E. R., On Dimensional Analysis and the Presentation of Data in Fluid Flow Problems, J. App. Mech, 1946, 68, A-34, March
- Perry, J. H. et al., "Standard System of Nomenclature for Chemical Engineering Unit Operations", Trans. Am. Inst. Chem. Engrs., 1944, 40, 251
- Moody, L. F., "Friction Factors for Pipe Flow", Trans. Am. Soc. Mech. Engrs., 1944, 66, 671
External links
- http://www.math.ntnu.no/~hanche/notes/buckingham/buckingham-a4.pdf
- http://www.knowledgedoor.com/1/Unit_Conversion/Dimensional_Analysis.htm
Category:Units of measure
Category:Procedural knowledge
-
ja:次元解析
Chemistry
Chemistry (derived from the Arabic word kimia, alchemy, where al is Arabic for the) is the science of matter that deals with the composition, structure, and properties of substances and with the transformations that they undergo. In the study of matter, chemistry also investigates its interactions with energy and itself (see physics, biology). Because of the diversity of matter, which is mostly composed of different combinations of atoms, chemists often study how atoms of different chemical elements interact to form molecules and how molecules interact with each other.
molecules
Introduction
Chemistry is a large field encompassing many subdisciplines that often overlap with significant portions of other sciences. The fundamental component of chemistry is that it involves matter in some way (this explains its broad reach). It may involve the interaction of matter with non-material phenomena such as energy. More central to chemistry is the interaction of matter with other matter such as in the classic chemical reaction where chemical bonds are broken and made, forming new molecules.
Matter, such as the chair you are sitting on or the air you breathe, is known today to consist of molecules. Each molecule consists of small bits of matter known as atoms that are connected together through chemical bonds. Each atom consists of smaller bits of matter known as subatomic particles. The structure of the world we commonly experience and the properties of the matter we commonly interact with are determined by the nature of this matter on the chemical level. Steel is hard because of how the atoms are bound together. Wood will burn because it can react with oxygen in a chemical reaction. Water is a liquid at room temperature because of how each molecule of water interacts with its neighbors. In fact, you are a thinking, sentient being because of an on-going series of chemical reactions and other chemical interactions. You can see this text because of how light interacts with molecules called proteins in the back of your eye.
Chemistry is often called the central science because it is what connects most of the other sciences together. Chemistry is in some ways physics on a larger scale and in some ways is biology or geology on a smaller scale. Chemistry is used to understand and make better materials for engineering. It is used to understand the chemical mechanisms of disease as well as to create pharmaceuticals to treat disease. Chemistry is somehow involved in almost every science, every technology and every "thing".
With such a large area of study, it is impossible to know everything about chemistry and very difficult to summarize the field concisely. Even the most knowledgable, experienced chemist only knows a very narrow area of chemistry better than others. Of course, most chemists have a broad general knowledge of many areas of chemistry as well. Chemistry is divided into many areas of study called subdisciplines in which chemists specialize. The chemistry taught at the high school or early college level is often called "general chemistry" and is intended to be an introduction to a wide variety of fundamental concepts and to give the student the tools to continue on to more advanced subjects. Many concepts presented at this level are often incomplete and technically inaccurate yet of extraordinary utility. Chemists regularly use these simple, elegant tools and explanations in their work when they suffice because the best solution possible is often so overwhelmingly difficult and the true solution is usually unobtainable.
The science of chemistry is historically a recent development but has its roots in alchemy which has been practiced for millennia throughout the world. The word chemistry is directly derived from the word alchemy, however the etymology of alchemy is unclear (see alchemy).
Subdisciplines of chemistry
Chemistry typically is divided into several major sub-disciplines. There are also several main cross-disciplinary and more specialized fields of chemistry.
; Analytical chemistry : Analytical chemistry is the analysis of material samples to gain an understanding of their chemical composition and structure. Analytical chemistry incorporates standardized experimental methods in chemistry. These methods may be used in all subdiciplines of chemistry, exluding purely theoretical chemistry.
; Biochemistry : Biochemistry is the study of the chemicals, chemical reactions and chemical interactions that take place in living organisms. Biochemistry and organic chemistry are closely related f.e. in medicinal chemistry.
; Inorganic chemistry : Inorganic chemistry is the study of the properties and reactions of inorganic compounds. The distinction between organic and inorganic disciplines is not absolute and there is much overlap, most importantly in the sub-discipline of organometallic chemistry.
; Organic chemistry : Organic chemistry is the study of the structure, properties, composition, mechanisms, and reactions of organic compounds.
; Physical chemistry : Physical chemistry or physicochemistry is the study of the physical basis of chemical systems and processes. In particular, the energetics and dynamics of such systems and processes are of interest to physical chemists. Important areas of study include chemical thermodynamics, chemical kinetics, electrochemistry, statistical mechanics, and spectroscopy. Physical chemistry has large overlap with molecular physics.
; Theoretical chemistry : Theoretical chemistry is the study of chemistry via theoretical reasoning (usually within mathematics or physics). In particular the application of quantum mechanics to chemistry is called quantum chemistry. Since the end of the second world war, the development of computers has allowed a systematic development of computational chemistry, which is the art of developing and applying computer programs for solving chemical problems. Theoretical chemistry has large overlap with molecular physics.
; Other fields : Astrochemistry, Atmospheric chemistry, Chemical Engineering, Electrochemistry, Environmental chemistry, Geochemistry, History of chemistry, Materials science, Medicinal chemistry, Molecular Biology, Molecular genetics, Nuclear chemistry, Organometallic chemistry, Petrochemistry, Pharmacology, Photochemistry, Phytochemistry, Polymer chemistry, Supramolecular chemistry, Surface chemistry, and Thermochemistry.
Fundamental concepts
Nomenclature
Nomenclature refers to the system for naming chemical compounds. There are well-defined systems in place for naming chemical species. Organic compounds are named according to the organic nomenclature system. Inorganic compounds are named according to the inorganic nomenclature system.
See also: IUPAC nomenclature
Atoms
Main article: Atom.
An atom is a collection of matter consisting of a positively charged core (the nucleus) which contains protons and neutrons, and which maintains a number of electrons to balance the positive charge in the nucleus.
Elements
Main article: Chemical element.
An element is a class of atoms which have the same number of protons in the nucleus. This number is known as the atomic number of the element. For example, all atoms with 6 protons in their nuclei are atoms of the chemical element carbon, and all atoms with 92 protons in their nuclei are atoms of the element uranium.
The most convenient presentation of the elements is in the periodic table, which groups elements with similar chemical properties together. Lists of the elements by name, by symbol, and by atomic number are also available.
See also: isotope
Compounds
Main article: Chemical compound
A compound is a substance with a fixed ratio of chemical elements which determines the composition, and a particular organisation which determines chemical properties. For example, water is a compound containing hydrogen and oxygen in the ratio of two to one, with the Oxygen between the hydrogens, and an angle of 104.5° between them. Compounds are formed and interconverted by chemical reactions.
Molecules
Main article: Molecule.
A molecule is the smallest indivisible portion of a pure compound that retains a set of unique chemical properties. A molecule consists of two or more atoms covalently bonded together.
Ions
Main article: Ion.
An ion is a charged species, or an atom or a molecule that has lost or gained an electron. Positively charged cations (e.g. sodium cation Na+) and negatively charged anions (e.g. chloride Cl-) can form neutral salts (e.g. sodium chloride NaCl). Examples of polyatomic ions that do not split up during acid-base reactions are hydroxide (OH-), or phosphate (PO43-).
Bonding
Main article: Chemical bond.
A chemical bond is an interaction which holds together atoms in molecules or crystals. In many simple compounds, valence bond theory and the concept of oxidation number can be used to predict molecular structure and composition. Similarly, theories from classical physics can be used to predict many ionic structures. With more complicated compounds, such as metal complexes, valence bond theory fails and alternative approaches which are based on quantum chemistry, such as molecular orbital theory, are necessary.
States of matter
Main article: Phase (matter).
A phase is a set of states of a chemical system that have similar bulk structural properties, over a range of conditions, such as pressure or temperature. Physical properties, such as density and refractive index tend to fall within values characteristic of the phase. The phase of matter is defined by the phase transition, which is when energy put into or taken out of the system goes into rearranging the structure of the system, instead of changing the bulk conditions.
Sometimes the distinction between phases can be continuous instead of having a discrete boundary, in this case the matter is considered to be in a supercritical state. When three states meet based on the conditions, it is known as a triple point and since this is invariant, it is a convenient way to define a set of conditions.
The most familiar examples of phases are solids, liquids, and gases. Less familiar phases include plasmas, Bose-Einstein condensates and fermionic condensates and the paramagnetic and ferromagnetic phases of magnetic materials. Even the familiar ice has many different phases, depending on the pressure and temperature of the system. While most familiar phases deal with three-dimensional systems, it is also possible to define analogs in two-dimensional systems, which is getting a lot of attention because of its relevance to biology.
Chemical reactions
Main article: Chemical reaction.
Chemical reactions are transformations in the fine structure of molecules. Such reactions can result in molecules attaching to each other to form larger molecules, molecules breaking apart to form two or more smaller molecules, or rearrangement of atoms within or across molecules. Chemical reactions usually involve the making or breaking of chemical bonds.
Quantum chemistry
Main article: Quantum chemistry.
Quantum chemistry describes the behavior of matter at the molecular scale. It is, in principle, possible to describe all chemical systems using this theory. In practice, only the simplest chemical systems may realistically be investigated in purely quantum mechanical terms, and approximations must be made for most practical purposes (e.g., Hartree-Fock, post Hartree-Fock or Density functional theory, see computational chemistry for more details). Hence a detailed understanding of quantum mechanics is not necessary for most chemistry, as the important implications of the theory (principally the orbital approximation) can be understood and applied in simpler terms.
Laws
The most fundamental concept in chemistry is the law of conservation of mass, which states that there is no detectable change in the quantity of matter during an ordinary chemical reaction. Modern physics shows that it is actually energy that is conserved, and that energy and mass are related; a concept which becomes important in nuclear chemistry. Conservation of energy leads to the important concepts of equilibrium, thermodynamics, and kinetics.
Further laws of chemistry elaborate on the law of conservation of mass. Joseph Proust's law of definite composition says that pure chemicals are composed of elements in a definite formulation; we now know that the structural arrangement of these elements is also important.
Dalton's law of multiple proportions says that these chemicals will present themselves in proportions that are small whole numbers (i.e. 1:2 O:H in water); although in many systems (notably biomacromolecules and minerals) the ratios tend to require large numbers, and are frequently represented as a fraction. Such compounds are known as Non-Stoichiometric Compounds
More modern laws of chemistry define the relationship between energy and transformations.
- In equilibrium, molecules exist in mixture defined by the transformations possible on the timescale of the equilibrium, and are in a ratio defined by the intrinsic energy of the molecules—the lower the intrinsic energy, the more abundant the molecule.
- Transforming one structure to another requires the input of energy to cross an energy barrier; this can come from the intrinsic energy of the molecules themselves, or from an external source which will generally accelerate transformations. The higher the energy barrier, the slower the transformation occurs.
- There is a hypothetical intermediate, or transition structure, that corresponds to the structure at the top of the energy barrier. The Hammond-Leffler Postulate states that this structure looks most similar to the product or starting material which has intrinsic energy closest to that of the energy barrier. Stabilizing this hypothetical intermediate through chemical interaction is one way to achieve catalysis.
- All chemical processes are reversible (law of microscopic reversibility) although some processes have such an energy bias, they are essentially irreversible.
History of chemistry
- Alchemy
- Discovery of the chemical elements
- History of chemistry
- Nobel Prize in chemistry
- Timeline of chemical element discovery
Etymology
Old French: alkemie; Arab al-kimia: the art of transformation. See also: alchemy
See also
- American Chemical Society
- Chemical engineering
- Chemist and list of chemists
- International Union of Pure and Applied Chemistry
- List of chemistry topics
- List of compounds
- List of important publications in chemistry
- Periodic table
- Chemistry resources
- Valency number
External links
- [http://www.allchemicals.info/ Chemical Glossary]
- [http://chem.sis.nlm.nih.gov/chemidplus/ Chemistry Information Database includes basic information and some toxicity]
- [http://www.chem.qmw.ac.uk/iupac/ IUPAC Nomenclature Home Page], see especially the "Gold Book" containing definitions of standard chemical terms
- [http://www.cci.ethz.ch/index.html Experiments] videos and photos of the techniques and results
- [http://physchem.ox.ac.uk/MSDS/ Material safety data sheets for a variety of chemicals]
- [http://www.flinnsci.com/search_MSDS.asp Material Safety Data Sheets]
Further reading
- Chang, Raymond. Chemistry 6th ed. Boston: James M. Smith, 1998. ISBN 0071152210.
Category:School subjects
als:Chemie
ko:화학
ms:Kimia
ja:化学
simple:Chemistry
th:เคมี
Base unitsThe SI system of units defines seven SI base units: fundamental physical units defined by an operational definition.
All other physical units can be derived from these base units: these are known as SI derived units. Derivation is by dimensional analysis. Use SI prefixes to abbreviate long numbers.
The following are the fundamental units from which all others are derived, they are dimensionally independent. The definitions stated below are widely accepted.
See also
- SI
- SI derived unit
- SI prefix
External links
- [http://www1.bipm.org/en/si/base_units/ BIPM]
- [http://www.npl.co.uk/mass/faqs/kilogram.html NPL - Kilogram]
- [http://physics.nist.gov/cuu/Units/index.html NIST -SI]
Category:SI units
Category:Dimensional analysis
ja:SI基本単位
ko:SI 기본 단위
simple:SI base unit
zh-cn:国际标准基准单位 zh-tw:國際標準基準單位
Speed:For alternate uses, see special education or speed (disambiguation).
Speed (symbol: v) is the rate of motion, or equivalently the rate of change of position, expressed as distance d moved per unit of time t.
Speed is a scalar quantity with dimensions distance/time; the equivalent vector quantity to speed is known as velocity. Speed is measured in the same physical units of measurement as velocity, but does not contain the element of direction that velocity has. Speed is thus the magnitude component of velocity.
Units of speed include:
- metres per second, (symbol m/s), the SI derived unit
- kilometres per hour, (symbol km/h)
- miles per hour, (symbol mph)
- knots (nautical miles per hour, symbol kt)
- Mach, where Mach 1 is the speed of sound; Mach n is n times as fast.
::Mach 1 = ~343 m/s = ~1235 km/h = ~768 mi/h (see the speed of sound for more detail)
- speed of light in vacuum (symbol c) is one of the natural units
::c = 299,792,458 m/s
- [other important conversions]
::1 m/s = 3.6 km/h
::1 mph = 1.609 km/h
::1 knot = 1.852 km/h = 0.514 m/s
Vehicles often have a speedometer to measure the speed.
The rate of change of speed with respect to time is termed acceleration.
Average speed
Speed as a physical property represents primarily instantaneous speed. In real life we often use average speed (denoted ), which is rate of total distance (or length) and time interval.
For example, if you go 60 miles in 2 hours, your average speed during that time is 60/2 = 30 miles per hour, but your instantaneous speed may have varied.
In mathematical notation:
:
Instantaneous speed defined as a function of time on interval gives average speed:
:
while instant speed defined as a function of distance (or length) on interval gives average speed:
:
It is often intuitively expected that going half a distance with speed and second half with speed , produce total average speed . The correct value is
(Note that the first is arithmetic mean while the second is harmonic mean).
Average speed can be derived also from speed distribution function (either in time or on distance):
:
:
Cultural significance
Speed or swiftness of motion plays a significant role in human culture, see racing. It is complementary to grace, precision and strength, e.g. in dancing or martial arts. Animals symbolizing speed are the horse (PIE
- ek'vos is etymologically derived from
- ok'u- "swift"), birds, especially raptors such as the hawk, and cats, e.g. the lynx (see e.g. Flos Duellatorum). The swiftest land animal is the cheetah, reaching running speeds of up to 110km/h.
See also
- Orders of magnitude (speed)
- Paul Virilio
External links
- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]
-
Category:Physical quantity
ko:속력
ja:速さ
nb:Fart
simple:Speed
th:ความเร็ว
Electric charge
Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interactions. Electrically charged matter is influenced by, and produces, electromagnetic fields. The interaction between charge and field is the source of one of the four fundamental forces, the electromagnetic force.
Overview
Electric charge is a characteristic of subatomic particles, and is quantized. For example, electrons have a charge, by convention, of −1. Protons have the opposite charge of +1. Quarks have a fractional charge of −1/3 or +2/3. The antiparticle equivalents of these have the opposite charge. There are other charged particles.
The SI unit of electric charge is the coulomb, which represents approximately 6.24 x 1018 elementary charges (the charge on a single electron or proton). The coulomb is defined as the quantity of charge that has passed through the cross-section of a conductor carrying one ampere within one second. The symbol Q is used to denote a quantity of electric charge.
Electric charge can be directly measured with an electrometer. The discrete nature of electric charge was demonstrated by Robert Millikan in his oil-drop experiment.
Formally, a measure of charge should be a multiple of the elementary charge e (charge is quantized), but since it is an average, macroscopic quantity, many orders of magnitude larger than a single elementary charge, it can effectively take on any real value.
History
As reported by the Ancient Greek philosopher Thales of Miletus around 600 BC, charge (or electricity) could be accumulated by rubbing fur on various substances, such as amber. The Greeks noted that the charged amber buttons could attract light objects such as hair. They also noted that if they rubbed the amber for long enough, they could even get a spark to jump. This property derives from the triboelectric effect. The word electricity derives from ηλεκτρον (electron), the Greek word for amber.
C. F. Du Fay proposed in 1733 [http://www.sparkmuseum.com/BOOK_DUFAY.HTM] that electricity came in two varieties which cancelled each other, and expressed this in terms of a two-fluid theory. When glass was rubbed with silk, DuFay said that the glass was charged with vitreous electricity, and when amber was rubbed with fur, the amber was said to be charged with resinous electricity.
By the 18th century, the study of electricity had become popular. One of the foremost experts was Benjamin Franklin, who argued in favor of a one-fluid theory of electricity. Franklin imagined electricity as being a type of invisible fluid present in all matter; for example he believed that it was the glass in a Leyden jar that held the accumulated charge. He posited that rubbing insulating surfaces together caused this fluid to change location, and that a flow of this fluid constitutes an electric current. He also posited that when matter contained too little of the fluid it was "negatively" charged, and when it had an excess it was "positively" charged. Arbitrarily (or for a reason that was not recorded) he identified the term "positive" with vitreous electricity and "negative" with resinous electricity. William Watson arrived at the same explanation at about the same time.
We now know that the Franklin/Watson model was close, but too simple. Matter is actually composed of several kinds of electrically charged particles, the most common being the positively charged proton and the negatively charged electron. Rather than one possible electric current there are many: a flow of electrons, a flow of electron "holes" which act like positive particles, or in electrolytic solutions, a flow of both negative and positive particles called ions moving in opposite directions. To reduce this complexity, electrical workers still use Franklin's convention and they imagine that electric current (known as conventional current) is a flow of exclusively positive particles. The conventional current simplifies electrical concepts and calculations, but it ignores the fact that within some conductors (electrolytes, semiconductors, and plasma), two or more species of electric charges flow in opposite directions. The flow direction for conventional current is also backwards compared to the actual electron drift taking place during electric currents in metals, the typical conductor of electricity, which is a source of confusion for beginners in electronics.
Properties
Aside from the properties described in articles about electromagnetism, charge is a relativistic invariant. This means that any particle that has charge q, no matter how fast it goes, always has charge q. This property has been experimentally verified by showing that the charge of one helium nucleus (two protons and two neutrons bound together in a nucleus and moving around at incredible speeds) is the same as two deuterium nuclei (one proton and one neutron bound together, but moving much more slowly than they would if they were in a helium nucleus).
Conservation of charge
The total electric charge of isolated systems remains constant regardless of changes within the system itself. This law is inherent to all processes known to physics and can be derived in a local form from Maxwell's equation as a continuity equation. More generally, the net change in charge density within a volume of integration is equal to the area integral over the current density on the surface of the volume , which is in turn equal to the net current :
:
See also
- Charge decay
- Current density
- Electrical discharge
- SI electromagnetism units
External links
- [http://www.unitconversion.org/unit_converter/charge.html Online Charge Converter] - convert between various units of charge, such as coulomb, EMU of charge, franklin, ampere-hour, faraday, and so on
- [http://www.unitconversion.org/unit_converter/charge-v.html Interactive Charge Conversion Table] - convert selected unit to all other units of charge
- [http://www.ce-mag.com/archive/2000/marapril/mrstatic.html How fast does a charge decay?]
-
Category:Electricity
Category:Physical quantity
Category:Chemical properties
Category:Introductory physics
Category:Fundamental physics concepts
ko:전하
ja:電荷
Units of measurementIntroduction
The definition, agreement and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day. Disparate systems of measurement used to be very common. Now there is a global standard, the International System (SI) of units, a form of metric system. The SI has been or is in the process of being adopted throughout the world. The United States of America is almost certainly the last to adopt the system but even there it is increasingly being used.
Standards are very important. Each unit is a set size. A distance or length or volume or mass or span of time being measured is described as a certain number of these units. A measurement may be quoted to a certain degree of accuracy.
One example of the importance of agreed units is the failure of the NASA Mars Climate Orbiter, which was accidentally destroyed on a mission to the planet Mars in September 1999 instead of entering orbit, due to miscommunications about the value of forces: different computer programs used different units of measurement (newton versus pound force). Enormous amounts of effort, time and money were wasted.
In physics and metrology units are standards for measurement of physical quantities that need clear definitions to be useful. Reproducibility of experimental results is central to the scientific method. A standard system of units facilitates this. Scientific systems of units are a refinement of the concept of weights and measures developed long ago for commercial purposes.
Science, medicine and engineering often use larger and smaller units of measurement than those used in day to day life and talk about them more exactly. The judicious selection of the units of measure can aid researchers in both framing and solving the problem.
History
Units of measurement were among the earliest tools invented by humans. Primitive societies needed rudimentary measures for many tasks: constructing dwellings of an appropriate size and shape, fashioning clothing, or bartering food or raw materials.
The earliest known uniform systems of weights and measures seem to have all been created sometime in the 4th and 3rd millennia BC among the ancient peoples of Mesopotamia, Egypt and the Indus Valley, and perhaps also Elam in Persia as well.
Many systems were based on the use of parts of the body and the natural surroundings as measuring instruments. Our present knowledge of early weights and measures comes from many sources.
Systems of measurement
A number of metric systems of units have evolved since the adoption of the original metric system in France in 1791. The current international standard metric system is the International system of units.
Prior to the global adoption of the metric system many different systems of measurement had been in use. Many of these were related to some extent or other. Often they were based on the dimensions of the human body.
Both the Imperial units and US customary units derive from earlier English units. Imperial units were mostly used in the British Commonwealth and the former British Empire. US customary units are the main system of measurement in the United States however some steps towards metrication have been made.
The above systems of units are based on arbitrary unit values, formalised as standards. Some unit values occur naturally in science. Systems of units based on these are called natural units. Similar to natural units, atomic units (au) are a convenient system of units of measurement used in atomic physics.
Also a great number of strange and non-standard units may be encountered. These may include: the ton of TNT, the Hiroshima atom bomb and the weight of an elephant.
Base and derived units
Different systems of units are based on different choices of a set of fundamental units.
The most widely used system of units is the International System of Units, or SI. There are seven SI base units. All other SI units can be derived from these base units.
For most quantities a unit is absolutely necessary to communicate values of that physical quantity. For example, conveying to someone a particular length without using some sort of unit is impossible, because a length cannot be described without a reference used to make sense of the value given.
But not all quantities require a unit of their own. Using physical laws, units of quantities can be expressed as combinations of units of other quantities. Thus only a small set of units is required. These units are taken as the base units. Other units are derived units. Derived units are a matter of convenience, as they can be expressed in terms of basic units. Which units are considered base units is a matter of choice.
The base units of SI are actually not the smallest set. Smaller sets have been defined. There are sets in which the electric and magnetic field have the same unit. This is based on physical laws that show that electric and magnetic field are actually different manifestations of the same phenomenon. In some fields of science such systems of units are highly favoured over the SI system.
Calculations with units
Units as dimensions
Any value of a physical quantity is expressed as a comparison to a unit of that quantity. For example, the value of a physical quantity Q is written as the product of a unit [Q] and a numerical factor:
:
The multiplication sign is usually left out, just as it is left out between variables in scientific notation of formulas. In formulas the unit [Q] can be treated as if it was a kind of physical dimension: see dimensional analysis for more on this treatment.
A distinction should be made between units and standards. A unit is fixed by its definition, and is independent of physical conditions such as temperature. By contrast, a standard is a physical realization of a unit, and realizes that unit only under certain physical conditions. For example, the metre is a unit, while a metal bar is a standard. One metre is the same length regardless of temperature, but a metal bar will be one metre long only at a certain temperature.
Guidelines
- Treat units like variables. Only add like terms. When a unit is divided by itself, the division yields a unitless one. When two different units are multiplied, the result is a new unit, referred to by the combination of the units. For instance, in SI, the unit of speed is metres per second (m/s). See dimensional analysis. A unit can be multiplied by itself, creating a unit with an exponent (e.g. m2/s2).
- Some units have special names, however these should be treated like their equivalents. For example, one newton (N) is equivalent to one kg m/s2. This creates the possiblity for units with multiple designations, for example: the unit for surface tension can be referred to as either N/m (newtons per metre) or kg/s2 (kilograms per second squared).
- Don't let definitions like density is mass per unit volume obscure your understanding of units. It sounds as if it says:
:D = m/m3 (mass divided by the unit of volume) (WRONG)
This is not true. The correct statement is that density is mass divided by volume:
:D = m/V (mass divided by volume, both variables)
Expressing a physical value in terms of another unit
Conversion of units involves comparison of different standard physical values, either of a single physical quantity or of a physical quantity and a combination of other physical quantities.
Starting with:
:
just replace the original unit [Q]_i with its meaning in terms of the desired unit , e.g. if , then:
:
Now and are both numerical values, so just calculate their product.
Or, which is just mathematically the same thing, multiply Q by unity, the product is still Q:
:
For example, you have an expression for a physical value Q involving the unit feet per second () and you want it in terms of the unit miles per hour ():
- Find facts relating the original unit to the desired unit:
:1 mile = 5280 feet and 1 hour = 3600 seconds
- Next use the above equations to construct a fraction that has a value of unity and that contains units such that, when it is multiplied with the original physical value, will cancel the original units:
:1 = (1 mile) / (5280 feet) and 1 = (3600 seconds) / (1 hour)
- Last, multiply the original expression of the physical value by the fraction, called a conversion factor, to obtain the same physical value expressed in terms of a different unit. Note: since the conversion factors have a numerical value of unity, multiplying any physical value by them will not change that value.
See also
- Metric system
- SI
- Natural units
- Planck units
- Geometrized units
- Atomic units
- Imperial unit
- US customary units
- List of strange units of measurement
- Conversion of units
- Historical weights and measures
- History of measurement
- International standard ISO 31: Quantities and units
- Units (computer program)
- Weights and measures
- Mesures usuelles
- Metrified English unit
- CODATA
- Metrication
- Metric system in the United States
- Metrology
- UTC (Coordinated Universal Time)
- Binary Prefixes - used to quantify large amounts of computer data
- Orders of magnitude
- ISO 31 - style guide for units of measurement
External links
- [http://www.eppo.go.th/ref/UNIT-OIL.html Unit Conversion : Oil Industry Conversions]
- [http://www.bipm.org/en/si Official SI website]
- [http://www.hmso.gov.uk/si/si1995/Uksi_19951804_en_2.htm UK - Units of Measurement Regulations 1995]
- [http://laws.justice.gc.ca/en/w-6/108519.html Canada - Weights and Measures Act 1970-71-72]
- [http://193.120.124.98/gen531996a.html Ireland - Metrology Act 1996]
- [http://www.ukma.org.uk UK Metric Association]
- [http://lamar.colostate.edu/~hillger US Metric Association]
- [http://aurora.rg.iupui.edu/UCUM/ucum.html The Unified Code for Units of Measure (UCUM)]
Converters
- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]
- [http://www.santos.com/ConversionCalculator.aspx?p=73 Santos - Conversion Calculator]
- [http://aurora.regenstrief.org/~schadow/units Regenstrief Unit Converter (for UCUM)]
References
Appendix B of NIST Handbook 44, 2002 Edition
Category:Measurement
-
-
Category:Systems of units
Category:Units of length
Category:Units of area
Category:Units of volume
Category:Units of mass
ja:物理単位
Conversion of unitsThis article lists conversion factors between a number of units of measurement.
Conversion techniques
The simplest way to convert from one unit to another is to carry through the units themselves in the mathematical operation. To illustrate this process, consider the following examples.
You would like to convert 6 feet into metres. Consulting the table below and finding that one foot is exactly 0.3048 metre, you can now perform the mathematical conversion:
6 ft × 0.3048 m/ft = 1.8288 m
Notice that the "foot" units canceled out, leaving only metres, the desired result. (Since 0.3048 metre per foot have infinite precision, the precision of the answer is determined by the precision of the 6 ft figure; if, for example, you are defining the fathom, expressing it with 5 significant figures is correct. But if the 6 ft figure is a measurement, the result needs to be rounded appropriately.)
Say your height is 183 centimetres, and you wish to convert this into inches:
183 cm / (2.54 cm/in) = 72.0 in
To check our answer, we convert this result back into feet:
72 in / (12 in/ft) = 6.0 feet
which confirms the earlier result.
Multiple units can be manipulated in the same fashion:
7 mi/s × 1.609344 km/mi × 3600 s/h = 40,000 km/h
Thus, Earth escape velocity is about 7 miles per second, or 40,000 kilometres per hour. Notice that since the calculation started with one significant figure (the 7), the answer also has one significant figure (the 4 in 40,000).
Deciding whether to multiply or divide is determined by looking at the units and deciding which ones you want to "get rid" of. In the conversion just above, if we had divided by 3600 s/h instead of multiplying, the result would have come out in kilometre-hours per square second, clearly an incorrect and meaningless result.
Rounding of results
An important thing to remember is that the process of making a conversion cannot give you any more precise results than what you started with. While many of the conversion factors given in the tables below are exact, and others while not exact contain many significant digits, all the numbers you get after performing calculations on a calculator or with pencil and paper are not meaningful.
After using these conversion factors, be sure to round off the results appropriately.
See also
- False precision
- Accuracy and precision
- Significant figures
Tables of conversion factors
Key:
≡ — definition
= — exactly equal to
≈ — approximately equal to
(digits) — indicates the digits repeat infinitely
Length
Area
Volume
Angle
Mass
|
| ≡ 60 kg
| = 60 kg
|-----
| quintal (metric) || q
|
| ≡ 100 kg
|-----
| wey ||
| ≡ 252 lb = 18 st
| = 114.305 277 24 kg (variants exist)
|-----
| long quarter||
| ≡ ¼ long tn
| = 254.011 727 2 kg
|-----
| kip || kip
| ≡ 1000 lb av
| = 453.592 37 kg
|-----
| short ton || sh tn
| ≡ 2000 lb
| = 907.184 74 kg
|-----
| tonne (mts unit) || t
|
| ≡ 1000 kg
|-----
| long ton|| long tn or ton
| ≡ 2240 lb
| = 1016.046 908 8 kg
|-----
| barge ||
| ≡ 22 ½ sh tn
| = 20,411.656 65 kg
|{| class="wikitable"
|+ Time, t
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-----
| second || s
|
| (SI base unit)
|-----
| Planck time ||
| ≡ √(Gℏ/c5)
| ≈ 1.351 211 818×10-43 s
|-----
| atomic unit of time || au
| ≡ a0/(α·c)
| ≈ 2.418 884 254×10-17 s
|-----
| svedberg || S
| ≡ 10-13 s
| = 100 fs
|-----
| shake ||
| ≡ 10-8 s
| = 10 ns
|-----
| sigma ||
| ≡ 10-6 s
| = 1 μs
|-----
| jiffy ||
| ≡ 1/60 s
| ≈ 16.666 667 ms
|-----
| jiffy (alternate) ||
| ≡ 1/100 s
| ≈ 10 ms
|-----
| helek ||
| ≡ 1/1080 h
| ≈ 3.333333 s
|-----
| minute || min
|
| ≡ 60 s
|-----
| hour || h
| ≡ 60 min
| = 3600 s
|-----
| day || d
| ≡ 24 h
| = 86 400 s
|-----
| week || wk
| ≡ 7 d
| = 604 800 s
|-----
| fortnight ||
| ≡ 2 wk
| = 1 209 600 s
|-----
| month (hollow) || mo
| ≡ 29 d
| = 2 505 600 s
|-----
| month (full) || mo
| ≡ 30 d
| = 2 592 000 s
|-----
| year (Calendar) || a or y
| ≡ 365 d
| = 31 536 000 s
|-----
| year (Gregorian) || a or y
| ≡ 365.2425 d
| = 31 556 952 s
|-----
| year (Julian) || a or y
| ≡ 365.25 d
| = 31 557 600 s
|-----
| sidereal year || a or y
| ≡ 365.256363 d
| = 31 558 149.76 s
|-----
| lustre; lustrum ||
| ≡ 5 a of 365 d
| = 1.5768×108 s
|-----
| octaeteris ||
| ≡ 8 a of 365 d
| = 2.522 88×108 s
|-----
| decade ||
| ≡ 10 a of 365 d
| = 3.1536×108 s
|-----
| enneadecaeteris; Metonic cycle ||
| ≡ 110 mo (hollow) + 125 mo (full) = 19 a of 365 d
| = 5.996 16×108 s
|-----
| Callippic cycle ||
| ≡ 441 mo (hollow) + 499 mo (full) = 76 a of 365.25 d
| = 2.398 377 6×109 s
|-----
| century (Calendar) ||
| ≡ 100 a of 365 d
| = 3.1536×109 s
|-----
| century (Julian) ||
| ≡ 100 a of 365.25 d
| = 3.155 76×109 s
|-----
| Hipparchic cycle ||
| ≡ 4 Callippic cycles - 1 d
| = 9.593 424×109 s
|-----
| millennium (Calendar) ||
| ≡ 1000 a of 365 d
| = 3.1536×1010 s
|-----
| millennium (Gregorian) ||
| ≡ 1000 a of 365.2425 d
| = 3.155 695 2×1010 s
|-----
| millennium (Julian) ||
| ≡ 1000 a of 365.25 d
| = 3.155 76×1010 s
|-----
| Sothic cycle ||
| ≡ 1461 a of 365 d
| = 4.607 409 6×1010 s
|{| class="wikitable"
|+ Speed, v
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-----
| metre per second (SI unit)|| m/s
|
| ≡ 1 m/s
|-----
| foot per hour || fph
| ≡ 1 ft/h
| ≈ 8.466 667×10-5 m/s
|-----
| furlong per fortnight ||
| ≡ ½ fur/wk
| ≈ 1.663 095×10-4 m/s
|-----
| inch per minute || ipm
| ≡ 1 in/min
| ≈ 4.23 333×10-4 m/s
|-----
| foot per minute || fpm
| ≡ 1 ft/min
| = 5.08×10-3 m/s
|-----
| inch per second || ips
| ≡ 1 in/s
| = 2.54×10-2 m/s
|-----
| kilometre per hour || km/h
| ≡ 1 km/h
| ≈ 2.777 778×10-1 m/s
|-----
| foot per second || fps
| ≡ 1 ft/s
| = 3.048×10-1 m/s
|-----
| mile per hour || mph
| ≡ 1 mi/h
| = 0.447 04 m/s
|-----
| knot || kn
| ≡ 1 NM/h = 1.852 km/h
| ≈ 0.514 444 m/s
|-----
| knot (Admiralty) || kn
| ≡ 1 NM (Adm)/h = 1.853 184 km/h
| ≈ 0.514 773 m/s
|-----
| mile per minute || mpm
| ≡ 1 mi/min
| = 26.8224 m/s
|-----
| mile per second || mps
| ≡ 1 mi/s
| = 1.609 344 km/s
|-----
| speed of light in vacuum || c
|
| ≡ 2.997 924 58×108 m/s
|{| class="wikitable"
|+ Acceleration, a
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-----
| metre per second squared (SI unit)|| m/s²
|
| ≡ 1 m/s²
|-----
| foot per hour per second || fph/s
| ≡ 1 ft/h·s
| ≈ 8.466 667×10-5 m/s²
|-----
| inch per minute per second || ipm/s
| ≡ 1 in/min·s
| ≈ 4.233 333×10-4 m/s²
|-----
| foot per minute per second || fpm/s
| ≡ 1 ft/min·s
| = 5.08×10-3 m/s²
|-----
| galileo || Gal
| ≡ 1 cm/s²
| = 10-2 m/s²
|-----
| inch per second squared || ips²
| ≡ 1 in/s²
| = 2.54×10-2 m/s²
|-----
| foot per second squared || fps²
| ≡ 1 ft/s²
| = 3.048×10-1 m/s²
|-----
| mile per hour per second || mph/s
| ≡ 1 mi/h·s
| = 4.4704×10-1 m/s²
|-----
| knot per second || kn/s
| ≡ 1 kn/s
| ≈ 5.144 444×10-1 m/s²
|-----
| standard gravity || g
|
| ≡ 9.806 65 m/s²
|-----
| mile per minute per second || mpm/s
| ≡ 1 mi/min·s
| = 26.8224 m/s²
|-----
| mile per second squared || mps²
| ≡ 1 mi/s²
| = 1.609 344×103 m/s²
|{| class="wikitable"
|+ Force, F
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-----
| newton (SI unit) || N
| ≡ kg·m/s²
|
|-----
| atomic unit of force || au
| ≡ me·α²·c²/a0
| ≈ 8.238 722 241×10-8 N
|-----
| dyne (cgs unit) || dyn
| ≡ g·cm/s²
| = 10-5 N
|-----
| gravet ||
| ≡ g × 1 g
| = 9.806 65 mN
|-----
| poundal || pdl
| ≡ 1 lb·ft/s²
| = 0.138 254 954 376 N
|-----
| ounce-force || ozf
| ≡ g × 1 oz
| = 0.278 013 850 953 781 2 N
|-----
| pound-force || lbf
| ≡ g × 1 lb
| = 4.448 221 615 260 5 N
|-----
| kilogram-force; kilopond; grave || kgf; kp
| ≡ g × 1 kg
| = 9.806 65 N
|-----
| sthene (mts unit) || sn
| ≡ 1 t·m/s²
| = 1 kN
|-----
| kip; kip-force || kip; kipf; klbf
| ≡ g × 1000 lb
| = 4.448 221 615 260 5 kN
|-----
| ton-force || tnf
| ≡ g × 1 sh tn
| = 8.896 443 230 521 kN |
|{| class="wikitable"
|+ Pressure, p (lower case)
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-----
| pascal (SI unit) || Pa
| ≡ N/m²
| = kg/m·s²
|-----
| barye (cgs unit) ||
| ≡ 1 dyn/cm²
| = 0.1 Pa
|-----
| poundal per square foot || pdl/sq ft
| ≡ 1 pdl/sq ft
| ≈ 1.488 164 Pa
|-----
| millimetre of water (3.98 °C) || mmH2O
| ≈ 999.972 kg/m³ × 1 mm × g
| = 9.806 38 Pa (= 0.999972 kgf/m²)
|-----
| pound per square foot || psf
| ≡ 1 lb/sq ft × g
| ≈ 47.880 259 Pa
|-----
| centimetre of water (3.98 °C) || cmH2O
| ≈ 999.972 kg/m³ × 1 cm × g
| = 98.0638 Pa
|-----
| torr || torr
| ≡ 101 325/760 Pa
| ≈ 133.322 368 4 Pa
|-----
| millimetre of mercury || mmHg
| ≡ 13 595.1 kg/m³ × 1 mm × g ≈ 1 torr
| = 133.322 387 415 Pa
|-----
| inch of water (3.98 °C) || inH2O
| ≈ 999.972 kg/m³ × 1 in × g
| = 249.082 Pa
|-----
| pièze (mts unit) || pz
| ≡ 1000 kg/m·s²
| = 1 kPa
|-----
| centimetre of mercury || cmHg
| ≡ 13 595.1 kg/m³ × 1 cm × g
| = 1.333 223 874 15 kPa
|-----
| foot of water (3.98 °C) || ftH2O
| ≈ 999.972 kg/m³ × 1 ft × g
| = 2.988 98 kPa
|-----
| inch of mercury || inHg
| ≡ 13 595.1 kg/m³ × 1 in × g
| = 3.386 388 640 341 kPa
|-----
| pound per square inch || psi
| ≡ 1 lb × g / 1 sq in
| ≈ 6.894 757×103 Pa
|-----
| foot of mercury || ftHg
| ≡ 13 595.1 kg/m³ × 1 ft × g
| = 40.636 663 684 091 9 kPa
|-----
| short ton per square foot ||
| ≡ 1 sh tn × g / 1 sq ft
| ≈ 95.760 518 kPa
|-----
| atmosphere (technical) || atm
| ≡ 1 kgf/cm²
| = 98.0665 kPa
|-----
| bar || bar
|
| ≡ 105 Pa
|-----
| atmosphere (standard) || atm
|
| ≡ 101 325 Pa
|-----
| kip per square inch || ksi
| ≡ 1 kipf/sq in
| ≈ 6.894757 MPa
|-----
| kilogram-force per square millimetre || kgf/mm²
| ≡ 1 kgf/mm²
| = 9.806 65 MPa
|{| class="wikitable"
|+ Energy, E, W
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-----
| joule (SI unit) || J
| ≡ N·m = W·s = V·A·s
| = kg·m²/s²
|-----
| electronvolt || eV
| ≡ e × 1 V
| ≈ 1.602 176×10-19 J
|-----
| rydberg || Ry
| ≡ R∞·ℎ·c
| ≈ 2.179 872×10-18 J
|-----
| hartree || Eh
| ≡ me·α²·c² (= 2 Ry)
| ≈ 4.359 744×10-18 J
|-----
| atomic unit of energy || au
| ≡ Eh ≈ 4.359 744×10-18 J
|-----
| erg (cgs unit) || erg
| ≡ 1 g·cm²/s²
| = 10-7 J
|-----
| foot-poundal || ft pdl
| ≡ 1 lb·ft²/s²
| = 4.214 011 009 380 48×10-2 J
|-----
| cubic centimetre of atmosphere; standard cubic centimetre || cc atm; scc
| ≡ 1 atm × 1 cm³
| = 0.101 325 J
|-----
| inch-pound force || in lbf
| ≡ g × 1 lb × 1 in
| = 0.112 984 829 027 616 7 J
|-----
| foot-pound force || ft lbf
| ≡ g × 1 lb × 1 ft
| = 1.355 817 948 331 400 4 J
|-----
| calorie (20 °C) || cal20 °C
|
| ≈ 4.1819 J
|-----
| calorie (thermochemical) || calth
|
| ≡ 4.184 J
|-----
| calorie (15 °C) || cal15 °C
|
| ≡ 4.1855 J
|-----
| calorie (International Table) || calIT
|
| ≡ 4.1868 J
|-----
| calorie (mean) || calmean
|
| ≈ 4.190 02 J
|-----
| calorie (3.98 °C) || cal3.98 °C
|
| ≈ 4.2045 J
|-----
| litre-atmosphere || l atm; sl
| ≡ 1 atm × 1 L
| = 101.325 J
|-----
| gallon-atmosphere (U.S.) || US gal atm
| ≡ 1 atm × 1 gal (US)
| = 383.556 849 013 8 J
|-----
| gallon-atmosphere (Imperial) || US gal atm
| ≡ 1 atm × 1 gal (Imp)
| = 460.632 569 25 J
|-----
| British thermal unit (thermochemical) || BTUth
| ≡ 1 lb/g × 1 calth × 1 °F/°C = 9.489 152 380 4 ÷ 9 kJ
| ≈ 1.054 350 kJ
|-----
| British thermal unit (ISO) || BTUISO
|
| ≡ 1.0545 kJ
|-----
| British thermal unit (63 °F) || BTU63 °F
|
| ≈ 1.0546 kJ
|-----
| British thermal unit (60 °F) || BTU60 °F
|
| ≈ 1.054 68 kJ
|-----
| British thermal unit (59 °F) || BTU59 °F
|
| ≡ 1.054 804 kJ
|-----
| British thermal unit (International Table) || BTUIT
| ≡ 1 lb/g × 1 calIT × 1 °F/°C
| = 1.055 055 852 62 kJ
|-----
| British thermal unit (mean) || BTUmean
|
| ≈ 1.055 87 kJ
|-----
| British thermal unit (39 °F) || BTU39 °F
|
| ≈ 1.059 67 kJ
|-----
| Celsius heat unit (International Table) || CHUIT
| ≡ 1 BTUIT × 1 °C/°F
| = 1.899 100 534 716 kJ
|-----
| cubic foot of atmosphere; standard cubic foot || cu ft atm; scf
| ≡ 1 atm × 1 ft³
| = 2.869 204 480 934 4 kJ
|-----
| kilocalorie; large calorie || kcal; Cal
| ≡ 1000 calIT
| = 4.1868 kJ
|-----
| cubic yard of atmosphere; standard cubic yard || cu yd atm; scy
| ≡ 1 atm × 1 yd³
| = 77.468 520 985 228 8 kJ
|-----
| cubic foot of natural gas ||
| ≡ 1000 BTUIT
| = 1.055 055 852 62 MJ
|-----
| horsepower-hour || hp·h
| ≡ 1 hp × 1 h
| = 2.6845 MJ
|-----
| Board of Trade Unit; kilowatt-hour || B.O.T.U.; kW·h
| ≡ 1 kW × 1 h
| = 3.6 MJ
|-----
| thermie || th
| ≡ 1 McalIT
| = 4.1868 MJ
|-----
| therm (U.S.) ||
| ≡ 100 000 BTU59 °F
| = 105.4804 MJ
|-----
| therm (E.C.) ||
| ≡ 100 000 BTUIT
| = 105.505 585 262 MJ
|-----
| ton of TNT || tTNT
| ≡ 1 Gcalth
| = 4.184 GJ
|-----
| barrel of oil equivalent || bboe
| ≈ 5.8 MBTU59 °F
| ≈ 6.12 GJ
|-----
| ton of coal equivalent || TCE
| ≡ 7 Gcalth
| = 29.3076 GJ
|-----
| ton of oil equivalent || TOE
| ≡ 10 Gcalth
| = 41.868 GJ
|-----
| quad ||
| ≡ 1015 BTUIT
| = 1.055 055 852 62 EJ
|{| class="wikitable"
|+ Power, P
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-----
| watt (SI unit) || W
| ≡ J/s = N·m/s
| = kg·m²/s³
|-----
| lusec || lusec
| ≡ 1 L·µmHg/s
| ≈ 1.333 224×10-4 W
|-----
| foot-pound-force per hour || ft lbf/h
| ≡ 1 ft lbf/h
| ≈ 3.766 161×10-4 W
|-----
| atmosphere cubic centimetre per minute || atm ccm
| ≡ 1 atm × 1 cm³/min
| = 1.688 75×10-3 W
|-----
| foot-pound-force per minute || ft lbf/min
| ≡ 1 ft lbf/min
| = 2.259 696 580 552 334×10-2 W
|-----
| atmosphere–cubic centimetre per second || atm ccs
| ≡ 1 atm × 1 cm³/s
| = 0.101 325 W
|-----
| BTU (International Table) per hour || BTUIT/h
| ≡ 1 BTUIT/h
| ≈ 0.293 071 W
|-----
| atmosphere–cubic foot per hour || atm cfh
| ≡ 1 atm × 1 cu ft/h
| = 0.797 001 244 704 W
|-----
| foot-pound-force per second || ft lbf/s
| ≡ 1 ft lbf/s
| = 1.355 817 948 331 400 4 W
|-----
| litre-atmosphere per minute || L·atm/min
| ≡ 1 atm × 1 L/min
| = 1.688 75 W
|-----
| calorie (International Table) per second || calIT/s
| ≡ 1 calIT/s
| = 4.1868 W
|-----
| BTU (International Table) per minute || BTUIT/min
| ≡ 1 BTUIT/min
| ≈ 17.584 264 W
|-----
| atmosphere-cubic foot per minute || atm·cfm
| ≡ 1 atm × 1 cu ft/min
| = 47.820 074 682 24 W
|-----
| square foot equivalent direct radiation || sq ft EDR
| ≡ 240 BTUIT/h
| ≈ 70.337 057 W
|-----
| litre-atmosphere per second || L·atm/s
| ≡ 1 atm × 1 L/s
| = 101.325 W
|-----
| horsepower (metric) || hp
| ≡ 75 m kgf/s
| = 735.498 75 W
|-----
| horsepower (European electrical) || hp
| ≡ 75 kp·m/s
| = 736 W
|-----
| horsepower (Imperial mechanical) || hp
| ≡ 550 ft lbf/s
| = 745.699 871 582 270 22 W
|-----
| horsepower (Imperial electrical) || hp
|
| ≡ 746 W
|-----
| ton of air conditioning ||
| ≡ 1 t × 1005 J/kg × 1 °F/K ÷ 10 min
| = 844.2 W
|-----
| poncelet || p
| ≡ 100 m kgf/s
| = 980.665 W
|-----
| BTU (International Table) per second || BTUIT/s
| ≡ 1 BTUIT/s
| = 1.055 055 852 62×10+3 W
|-----
| atmosphere-cubic foot per second || atm cfs
| ≡ 1 atm × 1 cu ft/s
| = 2.869 204 480 934 4×10+3 W
|-----
| ton of refrigeration (IT) ||
| ≡ 1 BTUIT × 1 sh tn/lb ÷ 10 min/s
| ≈ 3.516 853×10+3 W
|-----
| ton of refrigeration (Imperial) ||
| ≡ 1 BTUIT × 1 lng tn/lb ÷ 10 min/s
| ≈ 3.938 875×10+3 W
|-----
| boiler horsepower || bhp
| ≈ 34.5 lb/h × 970.3 BTUIT/lb
| ≈ 9.810 657×10+3 W
|{| class="wikitable"
|+ Action, Angular momentum, L, J
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-----
| SI unit || J·s
|
| ≡ kg·m²/s
|-----
| atomic unit of action || au
| ≡ ℏ = ℎ/2π
| ≈ 1.054 571 596×10-34 J·s
|-----
| cgs unit || erg·s
|
| = 10-7 J·s
|{| class="wikitable"
|+ Electric current, I
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-----
| ampere || A
|
| (SI base unit)
|-----
| esu per second; statampere (cgs unit) || esu/s
| ≡ (0.1 A·m/s)/c
| ≈ 3.335 641×10-10 A
|-----
| electromagnetic unit; abampere (cgs unit) || abamp
|
| ≡ 10 A
|{| class="wikitable"
|+ Electric charge, Q
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-----
| coulomb (SI unit) || C
|
| ≡ A·s
|-----
| atomic unit of charge || au
| ≡ e
| ≈ 1.602 176 462×10-19 C
|-----
| statcoulomb; franklin; electrostatic unit (cgs unit) || statC; Fr; esu
| ≡ (0.1 A·m)/c
| ≈ 3.335 641×10-10 C
|-----
| abcoulomb; electromagnetic unit (cgs unit) || abC; emu
|
| ≡ 10 C
|-----
| faraday || F
| ≡ 1 mol × NA·e
| ≈ 96,485.3383 C
|{| class="wikitable"
|+ Voltage, Electromotive force, U
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-----
| volt (SI unit) || V
|
| ≡ kg·m²/A·s³
|-----
| abvolt (cgs unit) || abV
|
| ≡ 1×10-8 V
|-----
| statvolt (cgs unit) || statV
| ≡ c·(1 μJ/A·m)
| = 299.792 458 V
|{| class="wikitable"
|+ Electrical resistance, R
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-----
| ohm (SI unit) || Ω
| ≡ V/A
| = kg·m²/A²s³
|{| class="wikitable"
|+ Dynamic viscosity, η
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-----
| pascal second (SI unit) || Pa·s
| ≡ N·s/m²
| = kg/m·s
|-----
| poise (cgs unit) || P
|
| ≡ 10-1 Pa·s
|{| class="wikitable"
|+ Kinematic viscosity, ν
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-----
| SI unit || m²/s
|
| ≡ m²/s
|-----
| stokes (cgs unit) || St
|
| ≡ 10-4 m²/s
|{| class="wikitable"
|+ Temperature, T
!Name of unit
!Symbol
!Relation to SI units
!Relation to °C
|-----
| kelvin || K
| SI base unit
| T[K] = T[°C] + 273.15
|-----
| degree Celsius || °C
| T[°C] = T[K] − 273.15
|
|-----
| degree Rankine || °R; °Ra
| T[°Ra] = 1.8 × T[K]
|
|-----
| degree Fahrenheit || °F
| T[°F] = T[K] × 1.8 − 459.67
| T[°F] = 1.8 × T[°C] + 32
|{
Exponential functionThe exponential function is one of the most important functions in mathematics. It is written as exp(x) or ex, where e is the base of the natural logarithm.
base of the natural logarithm
As a function of the real variable x, the graph of ex is always positive (above the x axis) and increasing (viewed left-to-right). It never touches the x axis, although it gets arbitrarily close to it (thus, the x axis is a horizontal asymptote to the graph). Its inverse function, the natural logarithm, ln(x), is defined for all positive x.
Sometimes, especially in the sciences, the term exponential function is reserved for functions of the form kax,
where a, called the base, is any positive real number. This article will focus initially on the exponential function with base e.
In general, the variable x can be any real or complex number, or even an entirely different kind of mathematical object; see the formal definition below.
Properties
Using the natural logarithm, one can define more general exponential functions. The function
:
defined for all a > 0, and all real numbers x, is called the exponential function with base a.
Note that the equation above holds for a = e, since
:
Exponential functions "translate between addition and multiplication" as is expressed in the following exponential laws:
:
:
:
:
:
:
These are valid for all positive real numbers a and b and all real numbers x and y. Expressions involving fractions and roots can often be simplified using exponential notation because:
:
and, for any a > 0, real number b, and integer n > 1:
:
For any real constant c holds:
:
for
Derivatives and differential equations
The importance of exponential functions in mathematics and the sciences stems mainly from properties of their derivatives. In particular,
:
That is, ex is its own derivative, a property unique among real-valued functions of a real variable. Other ways of saying the same thing include:
- The slope of the graph at any point is the height of the function at that point.
- The rate of increase of the function at x is equal to the value of the function at x.
- The function solves the differential equation .
- exp is a fixed point of derivative as a functional
In fact, many differential equations give rise to exponential functions, including the Schrödinger equation and the Laplace's equation as well as the equations for simple harmonic motion.
For exponential functions with other bases:
:
Thus any exponential function is a constant multiple of its own derivative.
If a variable's growth or decay rate is proportional to its size — as is the case in unlimited population growth (see Malthusian catastrophe), continuously compounded interest, or radioactive decay — then the variable can be written as a constant times an exponential function of time.
Furthermore for any differentiable function f(x) holds:
:
Formal definition
The exponential function ex can be defined in two equivalent ways, as an infinite series:
:
or as the limit of a sequence:
: .
In these definitions, stands for the factorial of n, and x can be any real number, complex number, element of a Banach algebra (for example, a square matrix), or member of the field of p-adic numbers.
For further explanation of these definitions and a proof of their equivalence, see the article Definitions of the exponential function.
Numerical value
To obtain the numerical value of the exponential function. The infinite series can be rewritten as :
:
:
This expression will converge quickly if we can ensure that x is less than one.
To ensure this, we can use the following identity.
:
- Where is the integer part of
- Where is the fractional part of
- Hence, is a always less than 1 and and add up to .
The value of the constant e^z can be calculated beforehand by multiplying e with itself z times.
On the complex plane
When considered as a function defined on the complex plane, the exponential function retains the important properties
:
:
:
:
for all z and w.
It is a holomorphic function which is periodic with imaginary period and can be written as
:
where a and b are real values. This formula connects the exponential function with the trigonometric functions and to the hyperbolic functions. Thus we see that all elementary functions except for the polynomials spring from the exponential function in one way or another.
See also Euler's formula.
Extending the natural logarithm to complex arguments yields a multi-valued function, ln(z). We can then define a more general exponentiation:
:
for all complex numbers z and w. This is also a multi-valued function. The above stated exponential laws remain true if interpreted properly as statements about multi-valued functions.
The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin. This can be seen by noting that the case of a line parallel with the real or imaginary axis maps to a line or circle.
Matrices and Banach algebras
The definition of the exponential function given above can be used verbatim for every Banach algebra, and in particular for square matrices (in which case the function is called the matrix exponential). In this case we have
:
:
: is invertible with inverse
: the derivative of at the point is that linear map which sends to .
In the context of non-commutative Banach algebras, such as algebras of matrices or operators on Banach or Hilbert spaces, the exponential function is often considered as a function of a real argument:
:
where A is a fixed element of the algebra and t is any real number. This function has the important properties
:
:
:
On Lie algebras
The "exponential map" sending a Lie algebra to the Lie group that gave rise to it shares the above properties, which explains the terminology. In fact, since R is the Lie algebra of the Lie group of all positive real numbers with multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie algebra M(n, R) of all square real matrices belongs to the Lie group of all invertible square matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.
Double exponential function
The term double exponential function can have two meanings:
- a function with two exponential terms, with different exponents
- a function f(x)=a^a^x; this grows even faster than an exponential function; for example, if a=10: f(-1)=1.26, f(0)=10, f(1)=1e10, f(2)=1e100=googol, f(3)=1e1000, ..., f(100)=googolplex.
Compare the super-exponential function, which grows even faster.
See also
- exponential growth
External links
-
Category:Special functions
Category:Complex analysis
Category:Exponentials
Category:Special hypergeometric functions
ko:지수함수
ja:指数関数
Logarithm
In mathematics, a logarithm is a function that gives the exponent in the equation bn = x. It is usually written as logb x = n. For example:
: thus | |
