# SI derived unit

Last updated

SI derived units are units of measurement derived from the seven SI base units specified by the International System of Units (SI). They can be expressed as a product (or ratio) of one or more of the base units, possibly scaled by an appropriate power of exponentiation (see: Buckingham π theorem). Some are dimensionless, as when the units cancel out in ratios of like quantities. SI coherent derived units involve only a trivial proportionality factor, not requiring conversion factors.

## Contents

The SI has special names for 22 of these coherent derived units (for example, hertz, the SI unit of measurement of frequency), but the rest merely reflect their derivation: for example, the square metre (m2), the SI derived unit of area; and the kilogram per cubic metre (kg/m3 or kg⋅m−3), the SI derived unit of density.

The names of SI coherent derived units, when written in full, are always in lowercase. However, the symbols for units named after persons are written with an uppercase initial letter. For example, the symbol for hertz is "Hz", while the symbol for metre is "m". [1]

## Special names

The International System of Units assigns special names to 22 derived units, which includes two dimensionless derived units, the radian (rad) and the steradian (sr).

Named units derived from SI base units [2]
Name Symbol Quantity Equivalents SI base unit
Equivalents
hertz Hz frequency 1/ss1
newton N force, weight kg⋅m/s2kg⋅m⋅s2
pascal Pa pressure, stress N/m2kg⋅m1⋅s2
joule J energy, work, heat m⋅N, C⋅V, W⋅skg⋅m2⋅s2
watt W power, radiant flux J/s, V⋅Akg⋅m2⋅s3
coulomb C electric charge or quantity of electricity s⋅A, F⋅Vs⋅A
volt V voltage, electrical potential difference, electromotive force W/A, J/Ckg⋅m2⋅s3⋅A1
farad F electrical capacitance C/V, s/Ωkg1⋅m2⋅s4⋅A2
ohm Ω electrical resistance, impedance, reactance 1/S, V/Akg⋅m2⋅s3⋅A2
siemens S electrical conductance 1/Ω, A/Vkg1⋅m2⋅s3⋅A2
weber Wb magnetic flux J/A, T⋅m2,V⋅skg⋅m2⋅s2⋅A1
tesla T magnetic induction, magnetic flux density V⋅s/m2, Wb/m2, N/(A⋅m)kg⋅s2⋅A1
henry H electrical inductance V⋅s/A, Ω⋅s, Wb/Akg⋅m2⋅s2⋅A2
degree Celsius °C temperature relative to 273.15 K KK
lumen lm luminous flux cd⋅srcd
lux lx illuminance lm/m2cd⋅m2
becquerel Bq radioactivity (decays per unit time)1/ss1
gray Gy absorbed dose (of ionizing radiation)J/kgm2⋅s2
sievert Sv equivalent dose (of ionizing radiation)J/kgm2⋅s2
katal kat catalytic activity mol/ss1⋅mol.

## By field of application

### Kinematics

NameSymbolQuantityExpression in terms
of SI base units
metre per second m/s speed, velocity m⋅s1
metre per second squared m/s2 acceleration m⋅s2
metre per second cubedm/s3 jerk, jolt m⋅s3
metre per second to the fourthm/s4 snap, jounce m⋅s4
hertz per secondHz/s frequency drift s2
cubic metre per second m3/s volumetric flow m3⋅s1

### Mechanics

NameSymbolQuantityExpression in terms
of SI base units
square metre m2 area m2
cubic metre m3 volume m3
newton-second N⋅s momentum, impulse m⋅kg⋅s1
newton metre secondN⋅m⋅s angular momentum m2⋅kg⋅s1
newton-metre N⋅m = J/rad torque, moment of force m2⋅kg⋅s2
newton per secondN/s yank m⋅kg⋅s3
reciprocal metre m1 wavenumber, optical power, curvature, spatial frequency m1
kilogram per square metrekg/m2 area density m2⋅kg
kilogram per cubic metre kg/m3 density, mass densitym3⋅kg
cubic metre per kilogramm3/kg specific volume m3⋅kg1
joule-second J⋅s action m2⋅kg⋅s1
joule per kilogramJ/kg specific energy m2⋅s2
joule per cubic metreJ/m3 energy density m1⋅kg⋅s2
newton per metreN/m = J/m2 surface tension, stiffness kg⋅s2
watt per square metreW/m2heat flux density, irradiance kg⋅s3
square metre per secondm2/s kinematic viscosity, thermal diffusivity, diffusion coefficientm2⋅s1
pascal-second Pa⋅s = N⋅s/m2dynamic viscosity m1⋅kg⋅s1
kilogram per metrekg/m linear mass density m1⋅kg
kilogram per secondkg/s mass flow rate kg⋅s1
watt per metreW/m spectral power m⋅kg⋅s3
gray per secondGy/s absorbed dose ratem2⋅s3
metre per cubic metrem/m3 fuel efficiency m2
watt per cubic metreW/m3 spectral irradiance, power density m1⋅kg⋅s3
joule per square metre secondJ/(m2⋅s) energy flux density kg⋅s3
reciprocal pascalPa1 compressibility m⋅kg1⋅s2
joule per square metreJ/m2 radiant exposure kg⋅s2
kilogram square metrekg⋅m2 moment of inertia m2⋅kg
newton metre second per kilogramN⋅m⋅s/kg specific angular momentum m2⋅s1
watt per steradian metreW/(sr⋅m) spectral intensity m⋅kg⋅s3

### Chemistry

NameSymbolQuantityExpression in terms
of SI base units
mole per cubic metremol/m3 molarity, amount of substance concentrationm3⋅mol
cubic metre per mole m3/mol molar volume m3⋅mol1
joule per kelvin moleJ/(K⋅mol) molar heat capacity, molar entropym2⋅kg⋅s2⋅K1⋅mol1
joule per mole J/molmolar energym2⋅kg⋅s2⋅mol1
siemens square metre per moleS⋅m2/mol molar conductivity kg−1⋅s3⋅A2⋅mol1
mole per kilogrammol/kg molality kg1⋅mol
kilogram per molekg/mol molar mass kg⋅mol1
cubic metre per mole secondm3/(mol⋅s) catalytic efficiency m3⋅s1⋅mol1

### Electromagnetics

NameSymbolQuantityExpression in terms
of SI base units
coulomb per square metreC/m2 electric displacement field, polarization density m2⋅s⋅A
coulomb per cubic metreC/m3electric charge density m3⋅s⋅A
ampere per square metreA/m2electric current density m2⋅A
siemens per metre S/m electrical conductivity m3⋅kg1⋅s3⋅A2
henry per metreH/m magnetic permeability m⋅kg⋅s2⋅A2
volt per metreV/m electric field strengthm⋅kg⋅s3⋅A1
ampere per metreA/m magnetization, magnetic field strengthm1⋅A
coulomb per kilogramC/kg exposure (X and gamma rays)kg1⋅s⋅A
ohm metre Ω⋅m resistivity m3⋅kg⋅s3⋅A2
coulomb per metreC/m linear charge density m1⋅s⋅A
joule per teslaJ/T magnetic dipole moment m2⋅A
square metre per volt secondm2/(V⋅s) electron mobility kg1⋅s2⋅A
reciprocal henryH1 magnetic reluctance m2⋅kg1⋅s2⋅A2
weber per metreWb/m magnetic vector potential m⋅kg⋅s2⋅A1
weber metreWb⋅m magnetic moment m3⋅kg⋅s2⋅A1
tesla metreT⋅m magnetic rigidity m⋅kg⋅s2⋅A1
metre per henrym/H magnetic susceptibility m1⋅kg1⋅s2⋅A2

### Photometry

NameSymbolQuantityExpression in terms
of SI base units
lumen second lm⋅s luminous energy s⋅cd
lux second lx⋅s luminous exposure m2⋅s⋅cd
candela per square metre cd/m2 luminance m2⋅cd
lumen per wattlm/W luminous efficacy m2⋅kg1⋅s3⋅cd

### Thermodynamics

NameSymbolQuantityExpression in terms
of SI base units
joule per kelvinJ/K heat capacity, entropy m2⋅kg⋅s2⋅K1
joule per kilogram kelvinJ/(K⋅kg) specific heat capacity, specific entropym2⋅s2⋅K1
watt per metre kelvinW/(m⋅K) thermal conductivity m⋅kg⋅s3⋅K1
kelvin per wattK/W thermal resistance m2⋅kg1⋅s3⋅K
reciprocal kelvinK1 thermal expansion coefficient K1
kelvin per metreK/m temperature gradient m1⋅K

## Other units used with SI

Some other units such as the hour, litre, tonne, bar, and electronvolt are not SI units, but are widely used in conjunction with SI units.

## Supplementary units

Until 1995, the SI classified the radian and the steradian as supplementary units, but this designation was abandoned and the units were grouped as derived units. [3]

## Related Research Articles

A physical quantity is a property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a value, which is the algebraic multiplication of a numerical value and a unit of measurement. For example, the physical quantity mass, symbol m, can be quantified as m=n kg, where n is the numerical value and kg is the unit symbol.

The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at the centre of a circle by an arc that is equal in length to the radius. The unit was formerly an SI supplementary unit and is currently a dimensionless SI derived unit, defined in the SI as 1 rad = 1 and expressed in terms of the SI base unit metre (m) as rad = m/m. Angles without explicitly specified units are generally assumed to be measured in radians, especially in mathematical writing.

The International System of Units, internationally known by the abbreviation SI, is the modern form of the metric system and the world's most widely used system of measurement. Established and maintained by the General Conference on Weights and Measures (CGPM), it is the only system of measurement with an official status in nearly every country in the world, employed in science, technology, industry, and everyday commerce.

A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The prefix kilo-, for example, may be added to gram to indicate multiplication by one thousand: one kilogram is equal to one thousand grams. The prefix milli-, likewise, may be added to metre to indicate division by one thousand; one millimetre is equal to one thousandth of a metre.

The steradian or square radian is the unit of solid angle in the International System of Units (SI). It is used in three dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radians, projected onto a circle, gives a length of a circular arc on the circumference, a solid angle in steradians, projected onto a sphere, gives the area of a spherical cap on the surface. The name is derived from the Greek στερεός stereos 'solid' + radian.

The mole (symbol mol) is the unit of measurement for amount of substance, a quantity proportional to the number of elementary entities of a substance. It is a base unit in the International System of Units (SI). One mole contains exactly 6.02214076×1023 elementary entities (602 sextillion or 602 billion times a trillion), which can be atoms, molecules, ions, or other particles. The number of particles in a mole is the Avogadro number (symbol N0) and the numerical value of the Avogadro constant (symbol NA) expressed in mol-1. The value was chosen based on the historical definition of the mole as the amount of substance that corresponds to the number of atoms in 12 grams of 12C, which made the mass of a mole of a compound expressed in grams numerically equal to the average molecular mass of the compound expressed in daltons. With the 2019 redefinition of the SI base units, the numerical equivalence is now only approximate but may be assumed for all practical purposes.

The caesium standard is a primary frequency standard in which the photon absorption by transitions between the two hyperfine ground states of caesium-133 atoms is used to control the output frequency. The first caesium clock was built by Louis Essen in 1955 at the National Physical Laboratory in the UK. and promoted worldwide by Gernot M. R. Winkler of the United States Naval Observatory.

The metric system is a system of measurement that succeeded the decimalised system based on the metre, which had been introduced in France in the 1790s. The historical development of these systems culminated in the definition of the International System of Units (SI) in the mid-20th century, under the oversight of an international standards body. Adopting the metric system is known as metrication.

A dimensionless quantity is a quantity to which no physical dimension is assigned. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Dimensionless quantities are distinct from quantities that have associated dimensions, such as time.

A base unit of measurement is a unit of measurement adopted for a base quantity. A base quantity is one of a conventionally chosen subset of physical quantities, where no quantity in the subset can be expressed in terms of the others. The SI base units, or Systeme International d'unites, consists of the metre, kilogram, second, ampere, kelvin, mole and candela.

A geometrized unit system, geometric unit system or geometrodynamic unit system is a system of natural units in which the base physical units are chosen so that the speed of light in vacuum, c, and the gravitational constant, G, are set equal to unity.

Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit of measurement. Mass, time, distance, heat, and angle are among the familiar examples of quantitative properties.

One turn is a unit of plane angle measurement equal to  radians, 360 degrees or 400 gradians. Thus it is the angular measure subtended by a complete circle at its center.

The radian per second is the unit of angular velocity in the International System of Units (SI). The radian per second is also the SI unit of angular frequency. The radian per second is defined as the angular frequency that results in the angular displacement increasing by one radian every second.

The International System of Quantities (ISQ) consists of the quantities used in physics and in modern science in general, starting with basic quantities such as length and mass, and the relationships between those quantities. This system underlies the International System of Units (SI) but does not itself determine the units of measurement used for the quantities.

The Unified Code for Units of Measure (UCUM) is a system of codes for unambiguously representing measurement units. Its primary purpose is machine-to-machine communication rather than communication between humans.

A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multiple of the unit of measurement.

Quantity calculus is the formal method for describing the mathematical relations between abstract physical quantities.

The history of the metric system began during the Age of Enlightenment with measures of length and weight derived from nature, along with their decimal multiples and fractions. The system became the standard of France and Europe within half a century. Other measures with unity ratios were added, and the system went on to be adopted across the world.

A coherent system of units is a system of units of measurement used to express physical quantities that are defined in such a way that the equations relating the numerical values expressed in the units of the system have exactly the same form, including numerical factors, as the corresponding equations directly relating the quantities. It is a system in which every quantity has a unique unit, or one that does not use conversion factors.

## References

1. Suplee, Curt (2 July 2009). "Special Publication 811". Nist.
2. International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), ISBN   92-822-2213-6, archived (PDF) from the original on 4 June 2021, retrieved 16 December 2021
3. "Resolution 8 of the CGPM at its 20th Meeting (1995)". Bureau International des Poids et Mesures . Retrieved 23 September 2014.

## Bibliography

• I. Mills, Tomislav Cvitas, Klaus Homann, Nikola Kallay, IUPAC (June 1993). Quantities, Units and Symbols in Physical Chemistry (2nd ed.). Blackwell Science Inc. p. 72.`{{cite book}}`: CS1 maint: multiple names: authors list (link)