SI derived unit explained

SI derived units are units of measurement derived from theseven 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.

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

Name	Symbol	Quantity	Equivalents	SI base unit Equivalents
hertz	Hz	frequency	1/s	s^−1
radian	rad	angle	m/m	1
steradian	sr	solid angle	m^2/m^2	1
newton	N	force, weight	kg⋅m/s^2	kg⋅m⋅s^−2
pascal	Pa	pressure, stress	N/m^2	kg⋅m^−1⋅s^−2
joule	J	energy, work, heat	m⋅N, C⋅V, W⋅s	kg⋅m^2⋅s^−2
watt	W	power, radiant flux	J/s, V⋅A	kg⋅m^2⋅s^−3
coulomb	C	electric charge or quantity of electricity	s⋅A, F⋅V	s⋅A
volt	V	voltage, electrical potential difference, electromotive force	W/A, J/C	kg⋅m^2⋅s^−3⋅A^−1
farad	F	electrical capacitance	C/V, s/Ω	kg^−1⋅m^−2⋅s^4⋅A^2
ohm	Ω	electrical resistance, impedance, reactance	1/S, V/A	kg⋅m^2⋅s^−3⋅A^−2
siemens	S	electrical conductance	1/Ω, A/V	kg^−1⋅m^−2⋅s^3⋅A^2
weber	Wb	magnetic flux	J/A, T⋅m^2,V⋅s	kg⋅m^2⋅s^−2⋅A^−1
tesla	T	magnetic induction, magnetic flux density	V⋅s/m^2, Wb/m^2, N/(A⋅m)	kg⋅s^−2⋅A^−1
henry	H	electrical inductance	V⋅s/A, Ω⋅s, Wb/A	kg⋅m^2⋅s^−2⋅A^−2
degree Celsius	°C	temperature relative to 273.15 K	K	K
lumen	lm		cd⋅sr	cd
lux	lx	illuminance	lm/m^2	cd⋅m^−2
becquerel	Bq	radioactivity (decays per unit time)	1/s	s^−1
gray	Gy	absorbed dose (of ionizing radiation)	J/kg	m^2⋅s^−2
sievert	Sv	equivalent dose (of ionizing radiation)	J/kg	m^2⋅s^−2
katal	kat	catalytic activity	mol/s	s^−1⋅mol.

By field of application

Kinematics

Name	Symbol	Quantity	Expression in terms of SI base units
	m/s		m⋅s^−1
	m/s^2		m⋅s^−2
metre per second cubed	m/s^3		m⋅s^−3
metre per second to the fourth	m/s^4		m⋅s^−4
	rad/s		s^−1
	rad/s^2		s^−2
hertz per second	Hz/s		s^−2
	m^3/s		m^3⋅s^−1

Mechanics

Name	Symbol	Quantity	Expression in terms of SI base units
	m^2		m^2
	m^3		m^3
	N⋅s		m⋅kg⋅s^−1
newton metre second	N⋅m⋅s		m^2⋅kg⋅s^−1
	N⋅m = J/rad		m^2⋅kg⋅s^−2
newton per second	N/s		m⋅kg⋅s^−3
	m^−1		m^−1
kilogram per square metre	kg/m^2		m^−2⋅kg
	kg/m^3	density, mass density	m^−3⋅kg
cubic metre per kilogram	m^3/kg		m^3⋅kg^−1
	J⋅s		m^2⋅kg⋅s^−1
joule per kilogram	J/kg		m^2⋅s^−2
joule per cubic metre	J/m^3		m^−1⋅kg⋅s^−2
newton per metre	N/m = J/m^2		kg⋅s^−2
watt per square metre	W/m^2		kg⋅s^−3
square metre per second	m^2/s	kinematic viscosity, thermal diffusivity, diffusion coefficient	m^2⋅s^−1
	Pa⋅s = N⋅s/m^2		m^−1⋅kg⋅s^−1
kilogram per metre	kg/m		m^−1⋅kg
kilogram per second	kg/s		kg⋅s^−1
watt per steradian square metre	W/(sr⋅m^2)		kg⋅s^−3
watt per steradian cubic metre	W/(sr⋅m^3)		m^−1⋅kg⋅s^−3
watt per metre	W/m		m⋅kg⋅s^−3
gray per second	Gy/s	absorbed dose rate	m^2⋅s^−3
metre per cubic metre	m/m^3		m^−2
watt per cubic metre	W/m^3		m^−1⋅kg⋅s^−3
joule per square metre second	J/(m^2⋅s)		kg⋅s^−3
reciprocal pascal	Pa^−1		m⋅kg^−1⋅s^2
joule per square metre	J/m^2		kg⋅s^−2
kilogram square metre	kg⋅m^2		m^2⋅kg
newton metre second per kilogram	N⋅m⋅s/kg		m^2⋅s^−1
watt per steradian	W/sr		m^2⋅kg⋅s^−3
watt per steradian metre	W/(sr⋅m)		m⋅kg⋅s^−3

Chemistry

Name	Symbol	Quantity	Expression in terms of SI base units
mole per cubic metre	mol/m^3	molarity, amount of substance concentration	m^−3⋅mol
	m^3/mol		m^3⋅mol^−1
joule per kelvin mole	J/(K⋅mol)	molar heat capacity, molar entropy	m^2⋅kg⋅s^−2⋅K^−1⋅mol^−1
	J/mol	molar energy	m^2⋅kg⋅s^−2⋅mol^−1
siemens square metre per mole	S⋅m^2/mol		kg^−1⋅s^3⋅A^2⋅mol^−1
mole per kilogram	mol/kg		kg^−1⋅mol
kilogram per mole	kg/mol		kg⋅mol^−1
cubic metre per mole second	m^3/(mol⋅s)		m^3⋅s^−1⋅mol^−1

Electromagnetics

Name	Symbol	Quantity	Expression in terms of SI base units
coulomb per square metre	C/m^2		m^−2⋅s⋅A
coulomb per cubic metre	C/m^3		m^−3⋅s⋅A
ampere per square metre	A/m^2		m^−2⋅A
	S/m	electrical conductivity	m^−3⋅kg^−1⋅s^3⋅A^2
farad per metre	F/m		m^−3⋅kg^−1⋅s^4⋅A^2
henry per metre	H/m		m⋅kg⋅s^−2⋅A^−2
volt per metre	V/m	electric field strength	m⋅kg⋅s^−3⋅A^−1
ampere per metre	A/m	magnetization, magnetic field strength	m^−1⋅A
coulomb per kilogram	C/kg	exposure (X and gamma rays)	kg^−1⋅s⋅A
	Ω⋅m		m^3⋅kg⋅s^−3⋅A^−2
coulomb per metre	C/m		m^−1⋅s⋅A
joule per tesla	J/T		m^2⋅A
square metre per volt second	m^2/(V⋅s)		kg^−1⋅s^2⋅A
reciprocal henry	H^−1		m^−2⋅kg^−1⋅s^2⋅A^2
weber per metre	Wb/m		m⋅kg⋅s^−2⋅A^−1
weber metre	Wb⋅m		m^3⋅kg⋅s^−2⋅A^−1
tesla metre	T⋅m		m⋅kg⋅s^−2⋅A^−1
ampere radian	A⋅rad		A
metre per henry	m/H		m^−1⋅kg^−1⋅s^2⋅A^2

Photometry

Name	Symbol	Quantity	Expression in terms of SI base units
	lm⋅s		s⋅cd
	lx⋅s		m^−2⋅s⋅cd
	cd/m^2		m^−2⋅cd
lumen per watt	lm/W		m^−2⋅kg^−1⋅s^3⋅cd

Thermodynamics

Name	Symbol	Quantity	Expression in terms of SI base units
joule per kelvin	J/K		m^2⋅kg⋅s^−2⋅K^−1
joule per kilogram kelvin	J/(K⋅kg)	specific heat capacity, specific entropy	m^2⋅s^−2⋅K^−1
watt per metre kelvin	W/(m⋅K)	thermal conductivity	m⋅kg⋅s^−3⋅K^−1
kelvin per watt	K/W		m^−2⋅kg^−1⋅s^3⋅K
reciprocal kelvin	K^−1		K^−1
kelvin per metre	K/m		m^−1⋅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.^[2]

See also

• Coherent derived unit
• International System of Quantities (ISQ)
• International Vocabulary of Metrology
• List of physical quantities
• Metric prefix
• Metric system

Bibliography

• Book: I. Mills, Tomislav Cvitas, Klaus Homann, Nikola Kallay, IUPAC. Quantities, Units and Symbols in Physical Chemistry. 2nd. June 1993. Blackwell Science Inc . 72.

Notes and References

1. Special Publication 811. Curt. Suplee. Nist . 2 July 2009.
2. Web site: Resolution 8 of the CGPM at its 20th Meeting (1995) . 2014-09-23 . Bureau International des Poids et Mesures.