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.[1][2] It is a system in which every quantity has a unique unit, or one that does not use conversion factors.[3]
A coherent derived unit is a derived unit that, for a given system of quantities and for a chosen set of base units, is a product of powers of base units, with the proportionality factor being one.[1]
If a system of quantities has equations that relate quantities and the associated system of units has corresponding base units, with only one unit for each base quantity, then it is coherent if and only if every derived unit of the system is coherent.
In the SI system, the derived unit m/s is a coherent derived unit for speed or velocity[4]:139 but km/h is not a coherent derived unit. Speed or velocity is defined by the change in distance divided by a change in time. The derived unit m/s uses the base units of the SI system.[1] The derived unit km/h requires numerical factors to relate to the SI base units: 1000m/km and 3600s/h.
In the cgs system, m/s is not a coherent derived unit. The numerical factor of 100cm/m is needed to express m/s in the cgs system.
History
Before the metric system
The earliest units of measure devised by humanity bore no relationship to each other.[citation needed] As both humanity's understanding of philosophical concepts and the organisation of society developed, so units of measurement were standardised – first particular units of measure had the same value across a community, then different units of the same quantity (for example feet and inches) were given a fixed relationship. Apart from Ancient China where the units of capacity and of mass were linked to red millet seed, there is little evidence of the linking of different quantities until the Enlightenment.[5]
Relating quantities of the same kind
The history of the measurement of length dates back to the early civilisations of the Middle East (10000BC – 8000BC). Archeologists have been able to reconstruct the units of measure in use in Mesopotamia, India, the Jewish culture and many others. Archeological and other evidence shows that in many civilisations, the ratios between different units for the same quantity of measure were adjusted so that they were integer numbers. In many early cultures such as Ancient Egypt, multiples of 2, 3 and 5 were not always used—the Egyptian royal cubit being 28 fingers or 7 hands.[6] In 2150BC, the Akkadian emperor Naram-Sin rationalised the Babylonian system of measure, adjusting the ratios of many units of measure to multiples of 2, 3 or 5, for example there were 6 she (barleycorns) in a shu-si (finger) and 30 shu-si in a kush (cubit).[7]
Measuring rod on exhibition in the Archeological Museum of Istanbul (Turkey) dating to the (3rd millennium BC) excavated at Nippur, Mesopotamia. The rod shows the various units of measure in use.
Relating quantities of different kinds
Non-commensurable quantities have different physical dimensions, which means that adding or subtracting them is not meaningful. For instance, adding the mass of an object to its volume has no physical meaning. However, new quantities (and, as such, units) can be derived via multiplication and exponentiation of other units. As an example, the SI unit for force is the newton, which is defined as kg⋅m⋅s−2. Since a coherent derived unit is one which is defined by means of multiplication and exponentiation of other units but not multiplied by any scaling factor other than 1, the pascal is a coherent unit of pressure (defined as kg⋅m−1⋅s−2), but the bar (defined as 100000kg⋅m−1⋅s−2) is not.
Note that coherence of a given unit depends on the definition of the base units. Should the standard unit of length change such that it is shorter by a factor of 100000, then the bar would be a coherent derived unit. However, a coherent unit remains coherent (and a non-coherent unit remains non-coherent) if the base units are redefined in terms of other units with the numerical factor always being unity.
Metric system
The concept of coherence was only introduced into the metric system in the third quarter of the nineteenth century; in its original form the metric system was non-coherent – in particular the litre was 0.001m3 and the are (from which we get the hectare) was 100m2. A precursor to the concept of coherence was however present in that the units of mass and length were related to each other through the physical properties of water, the gram having been designed as being the mass of one cubic centimetre of water at its freezing point.[8]
The CGS system had two units of energy, the erg that was related to mechanics and the calorie that was related to thermal energy, so only one of them (the erg, equivalent to the g⋅cm2/s2) could bear a coherent relationship to the base units. By contrast, coherence was a design aim of the SI, resulting in only one unit of energy being defined – the joule.[9]
Each variant of the metric system has a degree of coherence– the various derived units being directly related to the base units without the need of intermediate conversion factors.[1] An additional criterion is that, for example, in a coherent system the units of force, energy and power be chosen so that the equations
forceF = massm × accelerationa
energyE = forceF × distanced
powerP = energyE / timet
hold without the introduction of constant factors. Once a set of coherent units have been defined, other relationships in physics that use those units will automatically be true – Einstein's mass–energy equation, E=mc2, does not require extraneous constants when expressed in coherent units.[10]
Isaac Asimov wrote, "In the cgs system, a unit force is described as one that will produce an acceleration of 1 cm/sec2 on a mass of 1 gm. A unit force is therefore 1 cm/sec2 multiplied by 1 gm."[11] These are independent statements. The first is a definition; the second is not. The first implies that the constant of proportionality in the force law has a magnitude of one; the second implies that it is dimensionless. Asimov uses them both together to prove that it is the pure number one.
Asimov's conclusion is not the only possible one. In a system that uses the units foot (ft) for length, second (s) for time, pound (lb) for mass, and pound-force (lbf) for force, the law relating force (F), mass (m), and acceleration (a) is F = 0.031081 ma. Since the proportionality constant here is dimensionless and the units in any equation must balance without any numerical factor other than one, it follows that 1lbf = 1lb⋅ft/s2. This conclusion appears paradoxical from the point of view of competing systems, according to which F = ma and 1 lbf = 32.174 lb⋅ft/s2. Although the pound-force is a coherent derived unit in this system according to the official definition, the system itself is not considered to be coherent because of the presence of the proportionality constant in the force law.
A variant of this system applies the unit s2/ft to the proportionality constant. This has the effect of identifying the pound-force with the pound. The pound is then both a base unit of mass and a coherent derived unit of force. One may apply any unit one pleases to the proportionality constant. If one applies the unit s2/lb to it, then the foot becomes a unit of force. In a four-unit system (English engineering units), the pound and the pound-force are distinct base units, and the proportionality constant has the unit lbf⋅s2/(lb⋅ft).[12][13]
All these systems are coherent. One that is not is a three-unit system (also called English engineering units) in which F = ma that uses the pound and the pound-force, one of which is a base unit and the other, a noncoherent derived unit. In place of an explicit proportionality constant, this system uses conversion factors derived from the relation 1lbf = 32.174lb⋅ft/s2. In numerical calculations, it is indistinguishable from the four-unit system, since what is a proportionality constant in the latter is a conversion factor in the former. The relation among the numerical values of the quantities in the force law is {F} = 0.031081 {m} {a}, where the braces denote the numerical values of the enclosed quantities. Unlike in this system, in a coherent system, the relations among the numerical values of quantities are the same as the relations among the quantities themselves.
The following example concerns definitions of quantities and units. The (average) velocity (v) of an object is defined as the quantitative physical property of the object that is directly proportional to the distance (d) traveled by the object and inversely proportional to the time (t) of travel, i.e., v = kd/t, where k is a constant that depends on the units used. Suppose that the metre (m) and the second (s) are base units; then the kilometer (km) and the hour (h) are noncoherent derived units. The metre per second (mps) is defined as the velocity of an object that travels one metre in one second, and the kilometer per hour (kmph) is defined as the velocity of an object that travels one kilometre in one hour. Substituting from the definitions of the units into the defining equation of velocity we obtain, 1 mps = k m/s and 1 kmph = k km/h = 1/3.6 k m/s = 1/3.6 mps. Now choose k = 1; then the metre per second is a coherent derived unit, and the kilometre per hour is a noncoherent derived unit. Suppose that we choose to use the kilometre per hour as the unit of velocity in the system. Then the system becomes noncoherent, and the numerical value equation for velocity becomes {v} = 3.6 {d}/{t}. Coherence may be restored, without changing the units, by choosing k = 3.6; then the kilometre per hour is a coherent derived unit, with 1 kmph = 1m/s, and the metre per second is a noncoherent derived unit, with 1 mps = 3.6m/s.
A definition of a physical quantity is a statement that determines the ratio of any two instances of the quantity. The specification of the value of any constant factor is not a part of the definition since it does not affect the ratio. The definition of velocity above satisfies this requirement since it implies that v1/v2 = (d1/d2)/(t1/t2); thus if the ratios of distances and times are determined, then so is the ratio of velocities. A definition of a unit of a physical quantity is a statement that determines the ratio of any instance of the quantity to the unit. This ratio is the numerical value of the quantity or the number of units contained in the quantity. The definition of the metre per second above satisfies this requirement since it, together with the definition of velocity, implies that v/mps = (d/m)/(t/s); thus if the ratios of distance and time to their units are determined, then so is the ratio of velocity to its unit. The definition, by itself, is inadequate since it only determines the ratio in one specific case; it may be thought of as exhibiting a specimen of the unit.
A new coherent unit cannot be defined merely by expressing it algebraically in terms of already defined units. Thus the statement, "the metre per second equals one metre divided by one second", is not, by itself, a definition. It does not imply that a unit of velocity is being defined, and if that fact is added, it does not determine the magnitude of the unit, since that depends on the system of units. In order for it to become a proper definition both the quantity and the defining equation, including the value of any constant factor, must be specified. After a unit has been defined in this manner, however, it has a magnitude that is independent of any system of units.
List of coherent units
This list catalogues coherent relationships in various systems of units.
The centimetre–gram–second system of units is a variant of the metric system based on the centimetre as the unit of length, the gram as the unit of mass, and the second as the unit of time. All CGS mechanical units are unambiguously derived from these three base units, but there are several different ways in which the CGS system was extended to cover electromagnetism.
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.
The joule is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of one newton displaces a mass through a distance of one metre in the direction of that force. It is also the energy dissipated as heat when an electric current of one ampere passes through a resistance of one ohm for one second. It is named after the English physicist James Prescott Joule (1818–1889).
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. Coordinated by the International Bureau of Weights and Measures 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.
In science and engineering, the weight of an object, is the force acting on the object due to acceleration or gravity.
Specific impulse is a measure of how efficiently a reaction mass engine, such as a rocket using propellant or a jet engine using fuel, generates thrust. For engines like cold gas thrusters whose reaction mass is only the fuel they carry, specific impulse is exactly proportional to the effective exhaust gas velocity.
The metric system is a decimal-based system of measurement. The current international standard for the metric system is the International System of Units, in which all units can be expressed in terms of seven base units: the metre, kilogram, second, ampere, kelvin, mole, and candela.
The newton is the unit of force in the International System of Units (SI). It is defined as , the force which gives a mass of 1 kilogram an acceleration of 1 metre per second squared.
The atomic units are a system of natural units of measurement that is especially convenient for calculations in atomic physics and related scientific fields, such as computational chemistry and atomic spectroscopy. They were originally suggested and named by the physicist Douglas Hartree. Atomic units are often abbreviated "a.u." or "au", not to be confused with similar abbreviations used for astronomical units, arbitrary units, and absorbance units in other contexts.
A geometrized 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.
The poundal is a unit of force, introduced in 1877, that is part of the Absolute English system of units, which itself is a coherent subsystem of the foot–pound–second system.
Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units. The term "cgs units" is ambiguous and therefore to be avoided if possible: there are several variants of cgs with conflicting definitions of electromagnetic quantities and units.
Vacuum permittivity, commonly denoted ε0, is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric constant, or the distributed capacitance of the vacuum. It is an ideal (baseline) physical constant. Its CODATA value is:
Heaviside–Lorentz units constitute a system of units and quantities that extends the CGS with a particular set of equations that defines electromagnetic quantities, named for Oliver Heaviside and Hendrik Antoon Lorentz. They share with the CGS-Gaussian system that the electric constant ε0 and magnetic constant µ0 do not appear in the defining equations for electromagnetism, having been incorporated implicitly into the electromagnetic quantities. Heaviside–Lorentz units may be thought of as normalizing ε0 = 1 and µ0 = 1, while at the same time revising Maxwell's equations to use the speed of light c instead.
The foot–pound–second system is a system of units built on three fundamental units: the foot for length, the (avoirdupois) pound for either mass or force, and the second for time.
The ohm is the unit of electrical resistance in the International System of Units (SI). It is named after German physicist Georg Ohm. Various empirically derived standard units for electrical resistance were developed in connection with early telegraphy practice, and the British Association for the Advancement of Science proposed a unit derived from existing units of mass, length and time, and of a convenient scale for practical work as early as 1861.
The vacuum magnetic permeability, also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, conventionally written as μ0. Its purpose is to quantify the strength of the magnetic field emitted by an electric current. Expressed in terms of SI base units, it has the unit kg⋅m⋅s−2·A−2. It can be also expressed in terms of SI derived units, N·A−2.
The metre, kilogram, second system of units, also known more briefly as MKS units or the MKS system, is a physical system of measurement based on the metre, kilogram, and second (MKS) as base units. Distances are described in terms of metres, mass in terms of kilograms and time in seconds. Derived units are defined using the appropriate combinations, such as velocity in metres per second. Some units have their own names, such as the newton unit of force which is the combination kilogram metre per second squared.
A unit of measurement, or unit of measure, 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.
In particle physics and physical cosmology, Planck units are a system of units of measurement defined exclusively in terms of four universal physical constants: c, G, ħ, and kB. Expressing one of these physical constants in terms of Planck units yields a numerical value of 1. They are a system of natural units, defined using fundamental properties of nature rather than properties of a chosen prototype object. Originally proposed in 1899 by German physicist Max Planck, they are relevant in research on unified theories such as quantum gravity.
↑ McGreevy, Thomas (1995). Cunningham, Peter (ed.). The Basis of Measurement: Volume 1—Historical Aspects. Chippenham: Picton Publishing. Chapter 1: Some Ancient Units. ISBN0 948251 82 4.
↑ Asimov, Isaac (1966). Understanding Physics. New York: New American Library. Vol. I, p. 32.
↑ Comings, E. W. (1940). "English Engineering Units and Their Dimensions". Ind. Eng. Chem. 32 (7): 984–987. doi:10.1021/ie50367a028.
↑ Klinkenberg, Adrian (1969). "The American Engineering System of Units and Its Dimensional Constant gc". Ind. Eng. Chem. 61 (4): 53–59. doi:10.1021/ie50712a010.
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