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In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities and units of measurement and tracking these dimensions as calculations or comparisons are performed. The term dimensional analysis is also used to refer to conversion of units from one dimensional unit to another, which can be used to evaluate scientific formulae.
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.
A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is distinct from a mathematical constant, which has a fixed numerical value, but does not directly involve any physical measurement.
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 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 dalton or unified atomic mass unit is a non-SI unit of mass defined as 1/12 of the mass of an unbound neutral atom of carbon-12 in its nuclear and electronic ground state and at rest. The atomic mass constant, denoted mu, is defined identically, giving mu = 1/12 m(12C) = 1 Da.
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.
The Boltzmann constant is the proportionality factor that relates the average relative thermal energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, and in Planck's law of black-body radiation and Boltzmann's entropy formula, and is used in calculating thermal noise in resistors. The Boltzmann constant has dimensions of energy divided by temperature, the same as entropy. It is named after the Austrian scientist Ludwig Boltzmann.
In science and engineering, the parts-per notation is a set of pseudo-units to describe small values of miscellaneous dimensionless quantities, e.g. mole fraction or mass fraction. Since these fractions are quantity-per-quantity measures, they are pure numbers with no associated units of measurement. Commonly used are parts-per-million, parts-per-billion, parts-per-trillion and parts-per-quadrillion. This notation is not part of the International System of Units (SI) system and its meaning is ambiguous.
The becquerel is the unit of radioactivity in the International System of Units (SI). One becquerel is defined as the activity of a quantity of radioactive material in which one nucleus decays per second. For applications relating to human health this is a small quantity, and SI multiples of the unit are commonly used.
In physics, angular frequency, also called angular speed and angular rate, is a scalar measure of the angle rate or the temporal rate of change of the phase argument of a sinusoidal waveform or sine function . Angular frequency is the magnitude of the pseudovector quantity angular velocity.
Relative atomic mass, also known by the deprecated synonym atomic weight, is a dimensionless physical quantity defined as the ratio of the average mass of atoms of a chemical element in a given sample to the atomic mass constant. The atomic mass constant is defined as being 1/12 of the mass of a carbon-12 atom. Since both quantities in the ratio are masses, the resulting value is dimensionless. These definitions remain valid even after the 2019 redefinition of the SI base units.
IEC 60027 is a technical international standard for letter symbols published by the International Electrotechnical Commission (IEC), comprising the following parts:
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.
A Kibble balance is an electromechanical measuring instrument that measures the weight of a test object very precisely by the electric current and voltage needed to produce a compensating force. It is a metrological instrument that can realize the definition of the kilogram unit of mass based on fundamental constants.
ISO 31-0 is the introductory part of international standard ISO 31 on quantities and units. It provides guidelines for using physical quantities, quantity and unit symbols, and coherent unit systems, especially the SI. It was intended for use in all fields of science and technology and is augmented by more specialized conventions defined in other parts of the ISO 31 standard. ISO 31-0 was withdrawn on 17 November 2009. It is superseded by ISO 80000-1. Other parts of ISO 31 have also been withdrawn and replaced by parts of ISO 80000.
The joule-second is the unit of action and of angular momentum in the International System of Units (SI) equal to the product of an SI derived unit, the joule (J), and an SI base unit, the second (s). The joule-second is a unit of action or of angular momentum. The joule-second also appears in quantum mechanics within the definition of the Planck constant. Angular momentum is the product of an object's moment of inertia, in units of kg⋅m2 and its angular velocity in units of rad⋅s−1. This product of moment of inertia and angular velocity yields kg⋅m2⋅s−1 or the joule-second. The Planck constant represents the energy of a wave, in units of joule, divided by the frequency of that wave, in units of s−1. This quotient of energy and frequency also yields the joule-second (J⋅s).
In 2019, four of the seven SI base units specified in the International System of Quantities were redefined in terms of natural physical constants, rather than human artifacts such as the standard kilogram. Effective 20 May 2019, the 144th anniversary of the Metre Convention, the kilogram, ampere, kelvin, and mole are now defined by setting exact numerical values, when expressed in SI units, for the Planck constant, the elementary electric charge, the Boltzmann constant, and the Avogadro constant, respectively. The second, metre, and candela had previously been redefined using physical constants. The four new definitions aimed to improve the SI without changing the value of any units, ensuring continuity with existing measurements. In November 2018, the 26th General Conference on Weights and Measures (CGPM) unanimously approved these changes, which the International Committee for Weights and Measures (CIPM) had proposed earlier that year after determining that previously agreed conditions for the change had been met. These conditions were satisfied by a series of experiments that measured the constants to high accuracy relative to the old SI definitions, and were the culmination of decades of research.
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.
The Joint Committee for Guides in Metrology (JCGM) is an organization in Sèvres that prepared the Guide to the Expression of Uncertainty in Measurement (GUM) and the International Vocabulary of Metrology (VIM). The JCGM assumed responsibility for these two documents from the ISO Technical Advisory Group 4 (TAG4).
References
- 1 2 de Boer, J. (1995), "On the History of Quantity Calculus and the International System", Metrologia , 31 (6): 405–429, Bibcode:1995Metro..31..405D, doi:10.1088/0026-1394/31/6/001
- ↑ Fourier, Joseph (1822), Théorie analytique de la chaleur
- ↑ Maxwell, J. C. (1873), A Treatise on Electricity and Magnetism, Oxford: Oxford University Press, hdl: 2027/uc1.l0065867749
- ↑ Emerson, W.H. (2008), "On quantity calculus and units of measurement", Metrologia , 45 (2): 134–138, Bibcode:2008Metro..45..134E, doi:10.1088/0026-1394/45/2/002
- ↑ Johansson, I. (2010), "Metrological thinking needs the notions of parametric quantities, units and dimensions", Metrologia , 47 (3): 219–230, Bibcode:2010Metro..47..219J, doi:10.1088/0026-1394/47/3/012