Hilton has not yet commented on the minimum number of dimensionless physical constants from which all other dimensionless physical constants can be derived — but it can only be a matter of time.
|
|
Around the tray lie broken eggshells, cast off on a dimensionless blue surface.
|
|
The ratio is a dimensionless quantity without any physical or financial units because the barrels or dollars cancel each other out.
|
|
It looked more like a dimensionless plane of light than a wall, a field of illumination where the heavenly meets the earthly.
|
|
These threats include not only warming temperatures but also mutating viruses and political corruption and tend to be invisible, dimensionless and pervasive, like death.
|
|
Utopia can be the size of the world, a nation state, a skyscraper, a kiosk; it can an iPhone screen or the dimensionless, disembodied gamut of the internet.
|
|
Characters feel wooden and dimensionless (the conniving IT guy, the security guard with a heart of gold), and the production values come in just a hair above Syfy Movie quality.
|
|
The proposal is, of course, only viable if a universe that curves out of a dimensionless point in the way Hartle and Hawking imagined naturally grows into a universe like ours.
|
|
The "flat" trend governing the internet today has done well to eliminate the strange shadows of the skeuomorphic paradigm—in which the online world tried to mimic the look and feel of the real world—but it also gives us a dimensionless experience akin to shuffling through a deck of cards.
|
|
Justice is a singular noun but, in practice, true equality is more like a dimensionless mesh of justices: The right of the pussy to its health is not more important than the right of the transwoman to her life, nor is it more important than the right of the woman of color to a mainstream feminism that is not coded white by default.
|
|
Another important concept that can be applied towards the derivation the alternate depth equation arises from the comparison of the dimensionless momentum function to the dimensionless specific energy function. It can be seen that the dimensionless momentum function (M') has the identical functional relationship as the dimensionless specific energy function (E") when both are properly transformed. (Henderson 1966). From this comparison it can be observed that any result that applies to the dimensionless momentum equation (M') would likewise apply to the dimensionless specific energy equation (E").
|
|
All pure numbers are dimensionless quantities, for example 1, , , , and . Units of number such as the dozen, gross, googol, and Avogadro's number may also be considered dimensionless.
|
|
He argued that it also has the advantage of being dimensionless.
|
|
F is made to be dependent on the dimensionless quantity \,\\! \tau .
|
|
What is the period of oscillation of a mass attached to an ideal linear spring with spring constant suspended in gravity of strength ? That period is the solution for of some dimensionless equation in the variables , , , and . The four quantities have the following dimensions: [T]; [M]; [M/T2]; and [L/T2]. From these we can form only one dimensionless product of powers of our chosen variables, G_1 = T^2 k/m , and putting G_1 = C for some dimensionless constant gives the dimensionless equation sought.
|
|
The dimensionless product of powers of variables is sometimes referred to as a dimensionless group of variables; here the term "group" means "collection" rather than mathematical group. They are often called dimensionless numbers as well. Note that the variable does not occur in the group. It is easy to see that it is impossible to form a dimensionless product of powers that combines with , , and , because is the only quantity that involves the dimension L. This implies that in this problem the is irrelevant.
|
|
The thrust coefficient is another important dimensionless number in wind turbine aerodynamics.
|
|
That is, however, not the case here. When dimensional analysis yields only one dimensionless group, as here, there are no unknown functions, and the solution is said to be "complete" – although it still may involve unknown dimensionless constants, such as .
|
|
A g-factor (also called g value or dimensionless magnetic moment) is a dimensionless quantity that characterizes the magnetic moment and angular momentum of an atom, a particle or the nucleus. It is essentially a proportionality constant that relates the observed magnetic moment μ of a particle to its angular momentum quantum number and a unit of magnetic moment (to make it dimensionless), usually the Bohr magneton or nuclear magneton.
|
|
Bearing modulus is a modulus used in journal bearing design. It is a dimensionless number.
|
|
The urea reduction ratio (URR), is a dimensionless number used to quantify dialysis treatment adequacy.
|
|
Now the Mach number can be derived directly from the dimensionless form of the momentum equation.
|
|
Moritz Weber (1871 – 1951), was a professor of naval mechanics at the Polytechnic Institute of Berlin. The dimensionless numbers Reynolds number (named after the British scientist and mathematician Osborne Reynolds), and Froude number (named after the British engineer William Froude) was coined by Moritz Webber. Moreover, the dimensionless number Weber number was coined after him. Weber was also responsible in coining the term similitude to describe model studies that were scaled both geometrically and using dimensionless parameters for forces.
|
|
Thus, dimensional analysis may be used as a sanity check of physical equations: the two sides of any equation must be commensurable or have the same dimensions. This has the implication that most mathematical functions, particularly the transcendental functions, must have a dimensionless quantity, a pure number, as the argument and must return a dimensionless number as a result. This is clear because many transcendental functions can be expressed as an infinite power series with dimensionless coefficients.
|
|
It may be described in a very simple way by introducing a single dimensionless coupling parameter ʏ.
|
|
If differential operators are needed to describe the original system, their scaled counterparts become dimensionless differential operators.
|
|
By the rank–nullity theorem, a system of n vectors (matrix columns) in k linearly independent dimensions (the rank of the matrix is the number of fundamental dimensions) leaves a nullity, p, satisfying (p = n − k), where the nullity is the number of extraneous dimensions which may be chosen to be dimensionless. The dimensionless variables can always be taken to be integer combinations of the dimensional variables (by clearing denominators). There is mathematically no natural choice of dimensionless variables; some choices of dimensionless variables are more physically meaningful, and these are what are ideally used. The International System of Units defines k=7 base units, which are the ampere, kelvin, second, metre, kilogram, candela and mole.
|
|
If the forces in the Bond number are substituted with potential and cohesion energies, a new dimensionless number will be formed whereby the effect of the particles stiffness is also considered. This was firstly proposed by Behjani et al. where they introduced a dimensionless number titled as the Cohesion number.
|
|
Now the Rossby number can be derived directly from the dimensionless form of the momentum equation in curvilinear coordinates.
|
|
Dimensionless numbers in fluid mechanics are a set of dimensionless quantities that have an important role in analyzing the behavior of fluids. Common examples include the Reynolds or the Mach numbers, which describe as ratios the relative magnitude of fluid and physical system characteristics, such as density, viscosity, speed of sound, flow speed, etc.
|
|
Although the radian is a unit of measure, it is a dimensionless quantity. This can be seen from the definition given earlier: the angle subtended at the centre of a circle, measured in radians, is equal to the ratio of the length of the enclosed arc to the length of the circle's radius. Since the units of measurement cancel, this ratio is dimensionless. Although polar and spherical coordinates use radians to describe coordinates in two and three dimensions, the unit is derived from the radius coordinate, so the angle measure is still dimensionless.
|
|
If these values are used in dimensionless numbers, such as those listed below, much understanding of the phenomenon can be achieved.
|
|
The Weissenberg number (Wi) is a dimensionless number used in the study of viscoelastic flows. It is named after Karl Weissenberg. The dimensionless number compares the elastic forces to the viscous forces. It can be variously defined, but it is usually given by the relation of stress relaxation time of the fluid and a specific process time.
|
|
In general, the normalization factor can be difficult or impossible to compute, so the dimensionless quantities can be more useful in applications.
|
|
A drag count is a dimensionless unit used by aerospace engineers where 1 drag count is equal to a C_d of 0.0001.
|
|
Any ratio between physical constants of the same dimensions results in a dimensionless physical constant, for example, the proton-to- electron mass ratio. Any relation between physical quantities can be expressed as a relation between dimensionless ratios via a process known as nondimensionalisation. The term of "fundamental physical constant" is reserved for physical quantities which, according to the current state of knowledge, are regarded as immutable and as non-derivable from more fundamental principles. Notable examples are the speed of light c, and the gravitational constant G. The fine-structure constant α is the best known dimensionless fundamental physical constant.
|
|
The very same operational difference in measurement or perceived reality could just as well be caused by a change in h or e if α is changed and no other dimensionless constants are changed. It is only the dimensionless physical constants that ultimately matter in the definition of worlds.John Baez How Many Fundamental Constants Are There? This unvarying aspect of the Planck-relative scale, or that of any other system of natural units, leads many theorists to conclude that a hypothetical change in dimensionful physical constants can only be manifest as a change in dimensionless physical constants.
|
|
Another consequence of the theorem is that the functional dependence between a certain number (say, n) of variables can be reduced by the number (say, k) of independent dimensions occurring in those variables to give a set of p = n − k independent, dimensionless quantities. For the purposes of the experimenter, different systems that share the same description by dimensionless quantity are equivalent.
|
|
216, 218. It is often convenient to use the quantity \phi=f/P, the dimensionless fugacity coefficient, which is 1 for an ideal gas.
|
|
Specifically regarding VSL, if the SI meter definition was reverted to its pre-1960 definition as a length on a prototype bar (making it possible for the measure of c to change), then a conceivable change in c (the reciprocal of the amount of time taken for light to travel this prototype length) could be more fundamentally interpreted as a change in the dimensionless ratio of the meter prototype to the Planck length or as the dimensionless ratio of the SI second to the Planck time or a change in both. If the number of atoms making up the meter prototype remains unchanged (as it should for a stable prototype), then a perceived change in the value of c would be the consequence of the more fundamental change in the dimensionless ratio of the Planck length to the sizes of atoms or to the Bohr radius or, alternatively, as the dimensionless ratio of the Planck time to the period of a particular caesium-133 radiation or both.
|
|
The Buckingham theorem provides a method for computing sets of dimensionless parameters from given variables, even if the form of the equation remains unknown. However, the choice of dimensionless parameters is not unique; Buckingham's theorem only provides a way of generating sets of dimensionless parameters and does not indicate the most "physically meaningful". Two systems for which these parameters coincide are called similar (as with similar triangles, they differ only in scale); they are equivalent for the purposes of the equation, and the experimentalist who wants to determine the form of the equation can choose the most convenient one. Most importantly, Buckingham's theorem describes the relation between the number of variables and fundamental dimensions.
|
|
"Eurozone Officially Falls into Deflation, Piling Pressure on ECB." The Telegraph. Retrieved 27 December 2019. A percentage is a dimensionless number (pure number); it has no unit of measurement.
|
|
In fluid mechanics, the Reynolds number is the ratio of inertial forces (vsρ) to viscous forces (μ/L). It is one of the most important dimensionless numbers in fluid dynamics and is used, usually along with other dimensionless numbers, to provide a criterion for determining dynamic similitude. As such, the Reynolds number provides the link between modeling results (design) and the full-scale actual conditions. It can also be used to characterize the flow.
|
|
Dimensional Analysis and Self-Similarity Methods for Engineers and Scientists (2015). doi:10.1007/978-3-319-13476-5 To achieve kinematic similarity in a scaled model, dimensionless numbers in fluid dynamics come into consideration. For example, Reynolds number of the model and the prototype must match. There are other dimensionless numbers that will also come into consideration, such as Womersley numberLee Waite, Ph.D., P.E.; Jerry Fine, Ph.D.: Applied Biofluid Mechanics, Second Edition.
|
|
Since the kernel is only defined to within a multiplicative constant, if the above dimensionless constant is raised to any arbitrary power, it will yield another equivalent dimensionless constant. This example is easy because three of the dimensional quantities are fundamental units, so the last (g) is a combination of the previous. Note that if a2 were non-zero, there would be no way to cancel the M value; therefore a2 must be zero.
|
|
In this case the water+ice layer can be as thick as 250–300 km. Failing an ocean, the icy lithosphere may be somewhat thicker, up to about 300 km. Beneath the lithosphere and putative ocean, Callisto's interior appears to be neither entirely uniform nor particularly variable. Galileo orbiter data (especially the dimensionless moment of inertiaThe dimensionless moment of inertia referred to is I / (mr^2), where is the moment of inertia, the mass, and the maximal radius.
|
|
Paradoxically, dimensional analysis can be a useful tool even if all the parameters in the underlying theory are dimensionless, e.g., lattice models such as the Ising model can be used to study phase transitions and critical phenomena. Such models can be formulated in a purely dimensionless way. As we approach the critical point closer and closer, the distance over which the variables in the lattice model are correlated (the so-called correlation length, \xi ) becomes larger and larger.
|
|
The Dukhin number' (') is a dimensionless quantity that characterizes the contribution of the surface conductivity to various electrokinetic and electroacoustic effects, as well as to electrical conductivity and permittivity of fluid heterogeneous systems.
|
|
In the new SI system, the permeability of vacuum no longer has a defined value, but is a measured quantity, with an uncertainty related to that of the (measured) dimensionless fine structure constant.
|
|
Henri Coenraad Brinkman (Amsterdam, March 30, 1908 – Delft, February 1961) was a Dutch mathematician and physicist. He was a professor at the University of Groningen. The dimensionless Brinkman number is named after him.
|
|
In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned, also known as a bare, pure, or scalar quantity or a quantity of dimension one, with a corresponding unit of measurement in the SI of the unit one (or 1), which is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. An example of a quantity that has a dimension is time, measured in seconds.
|
|
In chemistry and physics, the dimensionless mixing ratio is the abundance of one component of a mixture relative to that of all other components. The term can refer either to mole ratio or mass ratio.
|
|
On 20 May 2019, a revision to the SI system went into effect, making the vacuum permeability no longer a constant but rather a value that needs to be determined experimentally; is a recently measured value in the new system. It is proportional to the dimensionless fine-structure constant with no other dependencies. A closely related property of materials is magnetic susceptibility, which is a dimensionless proportionality factor that indicates the degree of magnetization of a material in response to an applied magnetic field.
|
|
In other words, a scale invariant theory is one without any fixed length scale (or equivalently, mass scale) in the theory. For a scalar field theory with spacetime dimensions, the only dimensionless parameter satisfies = . For example, in = 4, only is classically dimensionless, and so the only classically scale-invariant scalar field theory in = 4 is the massless 4 theory. Classical scale invariance, however, normally does not imply quantum scale invariance, because of the renormalization group involved – see the discussion of the beta function below.
|
|
During the 1920s until his death, Eddington increasingly concentrated on what he called "fundamental theory" which was intended to be a unification of quantum theory, relativity, cosmology, and gravitation. At first he progressed along "traditional" lines, but turned increasingly to an almost numerological analysis of the dimensionless ratios of fundamental constants. His basic approach was to combine several fundamental constants in order to produce a dimensionless number. In many cases these would result in numbers close to 1040, its square, or its square root.
|
|
Mikhail Vasilyevich Ostrogradsky (transcribed also Ostrogradskiy, Ostrogradskiĭ) (, , September 24, 1801 – January 1, 1862) was a Russian.Mikhail Vasilyevich Ostrogradsky (Encyclopedia of Russian Academy of Sciences)Kunes, Josef. Dimensionless Physical Quantities in Science and Engineering. London — Waltham 2012.
|
|
For example, a concentration of 3% w/v usually means 3 g of substance in every 100 mL of solution. This cannot be converted to a dimensionless ratio, as in weight/weight or volume/volume fractions.
|
|
A key work he co-authored with co-worker Dr. Jerrard was A Dictionary of Scientific Units - Including dimensionless numbers and scales which ran to six editions between 1962 and 1993 and was translated into four languages.
|
|
Parts-per notations are all dimensionless quantities: in mathematical expressions, the units of measurement always cancel. In fractions like "2 nanometers per meter" (2 n ~~m~~ / ~~m~~ = 2 nano = 2 × 10−9 = 2 ppb = 2 × ), so the quotients are pure-number coefficients with positive values less than or equal to 1\. When parts-per notations, including the percent symbol (%), are used in regular prose (as opposed to mathematical expressions), they are still pure-number dimensionless quantities. However, they generally take the literal "parts per" meaning of a comparative ratio (e.g.
|
|
In physics, the proton-to-electron mass ratio, μ or β, is simply the rest mass of the proton (a baryon found in atoms) divided by that of the electron (a lepton found in atoms). Because this is a ratio of like-dimensioned physical quantities, it is a dimensionless quantity, a function of the dimensionless physical constants, and has numerical value independent of the system of units, namely: :μ = The number enclosed in parentheses is the measurement uncertainty on the last two digits. The value of μ is known to about 0.1 parts per billion.
|
|
By the 1940s, it became clear that the value of the fine-structure constant deviates significantly from the precise value of 1/137, refuting Eddington's argument. With the development of quantum chemistry in the 20th century, however, a vast number of previously inexplicable dimensionless physical constants were successfully computed from theory. In light of that, some theoretical physicists still hope for continued progress in explaining the values of other dimensionless physical constants. It is known that the Universe would be very different if these constants took values significantly different from those we observe.
|
|
The Buckingham theorem indicates that validity of the laws of physics does not depend on a specific unit system. A statement of this theorem is that any physical law can be expressed as an identity involving only dimensionless combinations (ratios or products) of the variables linked by the law (e. g., pressure and volume are linked by Boyle's law – they are inversely proportional). If the dimensionless combinations' values changed with the systems of units, then the equation would not be an identity, and Buckingham's theorem would not hold.
|
|
Quantities having dimension 1, dimensionless quantities, regularly occur in sciences, and are formally treated within the field of dimensional analysis. In the nineteenth century, French mathematician Joseph Fourier and Scottish physicist James Clerk Maxwell led significant developments in the modern concepts of dimension and unit. Later work by British physicists Osborne Reynolds and Lord Rayleigh contributed to the understanding of dimensionless numbers in physics. Building on Rayleigh's method of dimensional analysis, Edgar Buckingham proved the theorem (independently of French mathematician Joseph Bertrand's previous work) to formalize the nature of these quantities.
|
|
The Buckingham theorem indicates that validity of the laws of physics does not depend on a specific unit system. A statement of this theorem is that any physical law can be expressed as an identity involving only dimensionless combinations (ratios or products) of the variables linked by the law (e. g., pressure and volume are linked by Boyle's Law – they are inversely proportional). If the dimensionless combinations' values changed with the systems of units, then the equation would not be an identity, and Buckingham's theorem would not hold.
|
|
In convective heat transfer, the Churchill–Bernstein equation is used to estimate the surface averaged Nusselt number for a cylinder in cross flow at various velocities. The need for the equation arises from the inability to solve the Navier–Stokes equations in the turbulent flow regime, even for a Newtonian fluid. When the concentration and temperature profiles are independent of one another, the mass-heat transfer analogy can be employed. In the mass-heat transfer analogy, heat transfer dimensionless quantities are replaced with analogous mass transfer dimensionless quantities.
|
|
The cosmological constant, which can be thought of as the density of dark energy in the universe, is a fundamental constant in physical cosmology that has a dimensionless value of approximately 10−122.Jaffe, R. L., & Taylor, W., The Physics of Energy (Cambridge: Cambridge University Press, 2018), p. 419. Other dimensionless constants are the measure of homogeneity in the universe, denoted by Q, which is explained below by Martin Rees, the baryon mass per photon, the cold dark matter mass per photon and the neutrino mass per photon.
|
|
In particle physics, dimensional transmutation is a physical mechanism providing a linkage between a dimensionless parameter and a dimensionful parameter. In classical field theory, such as gauge theory in four-dimensional spacetime, the coupling constant is a dimensionless constant. However, upon quantization, logarithmic divergences in one-loop diagrams of perturbation theory imply that this "constant" actually depends on the typical energy scale of the processes under considerations, called the renormalization group (RG) scale. This "running" of the coupling is specified by the beta-function of the renormalization group.
|
|
The strain is the ratio of two lengths, so it is a dimensionless quantity (a number that does not depend on the choice of measurement units). Thus, strain rate is in units of inverse time (such as s−1).
|
|
Typical chemical shifts are rarely more than a few hundred Hz from the reference frequency, so chemical shifts are conveniently expressed in ppm (Hz/MHz). Parts-per notation gives a dimensionless quantity that does not depend on the instrument's field strength.
|
|
Automotive aerodynamics of motorcycles are not as frequently measured by third parties or reported by manufacturers as other figures like power. The dimensionless measure of drag coefficient, Cd, varies from .55 to .65 (comparable to a pickup truck), vs Cd .
|
|
For example, a few percent change in the value of the fine structure constant would be enough to eliminate stars like our Sun. This has prompted attempts at anthropic explanations of the values of some of the dimensionless fundamental physical constants.
|
|
Thrust-to-weight ratio is, as its name suggests, the ratio of instantaneous thrust to weight (where weight means weight at the Earth's standard acceleration g_0).Sutton and Biblarz 2000, p. 442. Quote: "thrust-to-weight ratio F/W0 is a dimensionless parameter that is identical to the acceleration of the rocket propulsion system (expressed in multiples of g0) if it could fly by itself in a gravity free vacuum." It is a dimensionless parameter characteristic of rockets and other jet engines and of vehicles propelled by such engines (typically space launch vehicles and jet aircraft).
|
|
The Buckingham π theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of dimensionless parameters, where m is the rank of the dimensional matrix. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables. A dimensional equation can have the dimensions reduced or eliminated through nondimensionalization, which begins with dimensional analysis, and involves scaling quantities by characteristic units of a system or natural units of nature. This gives insight into the fundamental properties of the system, as illustrated in the examples below.
|
|
Eq. 5 cannot apply for large sizes because it approaches for D \to \infty a horizontal asymptote. For large sizes, \sigma_N must approach the Weibull statistical size effect, Eq. 3. This condition is satisfied by the generalized energetic-statistical size effect law: where r, m are empirical constants (r n_d/m < 1). The deterministic formula (5) is recovered as the limit case for m \rightarrow \infty. (Fig. 2d) shows a comparison of the last formula with the test results for many different concretes, plotted as dimensionless strength \sigma_N /f'_t versus dimensionless structure size D/D_0.
|
|
Consequently, the interaction may be characterised by a dimensionful parameter , namely the value of the RG scale at which the coupling constant diverges. In the case of quantum chromodynamics, this energy scale is called the QCD scale, and its value 220 MeV supplants the role of the original dimensionless coupling constant in the form of the logarithm (at one- loop) of the ratio and . Perturbation theory, which produced this type of running formula, is only valid for a (dimensionless) coupling ≪ 1. In the case of QCD, the energy scale is an infrared cutoff, such that implies , with the RG scale.
|
|
Yet, maintaining some properties at their natural values can lead to erroneous predictions. Properties such as viscosity, friction, and surface area must be adjusted to maintain appropriate flow and transport behavior. This usually involves matching dimensionless ratios (e.g., Reynolds number, Froude number).
|
|
In Karlsruhe (Germany) on October 26, 1896 a memorial was inaugurated to honor his efforts. The Grashof Number was named after him. It is a very important dimensionless parameter in analyzing natural or free convection. The Grashof Condition was also named after him.
|
|
Being a ratio the FCR is dimensionless, that is, it is not affected by the units of measurement used to determine the FCR.Stickney, Robert R. (2009) Aquaculture: An Introductory Text, page 248, CABI, . FCR a function of the animal's geneticsArthur P.F. et al.
|
|
Prevalence, a common measure in epidemiology is strictly a type of denominator data, a dimensionless ratio or proportion. Prevalence may be expressed as a fraction, a percentage or as the number of cases per 1,000, 10,000 or 100,000 in the population of interest.
|
|
This metric has only two undetermined parameters. An overall dimensionless length scale factor R describes the size scale of the universe as a function of time; an increase in R is the expansion of the universe. A curvature index k describes the geometry.
|
|
For a 1000-pound vehicle, it would take 1000 times more tow force, i.e. 10 pounds. One could say that C_{rr} is in lb(tow-force)/lb(vehicle weight). Since this lb/lb is force divided by force, C_{rr} is dimensionless.
|
|
Strouhal's major contribution to the fundamentals of fluid mechanics was his discovery in 1878 of the Strouhal number (St). This dimensionless number describing oscillating flow mechanisms was discovered by Strouhal while experimenting in 1878 with wires experiencing vortex shedding and singing in the wind.
|
|
If the numerical values of the dimensional quantities change, but corresponding dimensionless quantities remain invariant then we can argue that snapshots of the system at different times are similar. When this happens we say that the system is self-similar. One way of verifying dynamic scaling is to plot dimensionless variables f/t^\theta as a function of x/t^z of the data extracted at various different time. Then if all the plots of f vs x obtained at different times collapse onto a single universal curve then it is said that the systems at different time are similar and it obeys dynamic scaling.
|
|
The dimensionless heat capacity divided by three, as a function of temperature as predicted by the Debye model and by Einstein's earlier model. The horizontal axis is the temperature divided by the Debye temperature. Note that, as expected, the dimensionless heat capacity is zero at absolute zero, and rises to a value of three as the temperature becomes much larger than the Debye temperature. The red line corresponds to the classical limit of the Dulong–Petit law In most solids (but not all), the molecules have a fixed mean position and orientation, and therefore the only degrees of freedom available are the vibrations of the atoms.
|
|
In physics, the fine-structure constant, also known as Sommerfeld's constant, commonly denoted by (the Greek letter alpha), is a fundamental physical constant characterizing the strength of the electromagnetic interaction between elementary charged particles. It is a dimensionless quantity related to the elementary charge , which characterizes the strength of the coupling of an elementary charged particle with the electromagnetic field, by the formula . As a dimensionless quantity, its numerical value, approximately , is independent of the system of units used. While there are multiple physical interpretations for , it received its name from Arnold Sommerfeld, who introduced it in 1916, when extending the Bohr model of the atom.
|
|
Materials that are not permanent magnets usually satisfy the relation M = χH in SI, where χ is the (dimensionless) magnetic susceptibility. Most non-magnetic materials have a relatively small χ (on the order of a millionth), but soft magnets can have χ on the order of hundreds or thousands. For materials satisfying M = χH, we can also write B = μ0(1 + χ)H = μ0μrH = μH, where μr = 1 + χ is the (dimensionless) relative permeability and μ =μ0μr is the magnetic permeability. Both hard and soft magnets have a more complex, history-dependent, behavior described by what are called hysteresis loops, which give either B vs.
|
|
Relative atomic mass (symbol: A) or 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 (symbol: m) is defined as being of the mass of a carbon-12 atom. Since both quantities in the ratio are masses, the resulting value is dimensionless; hence the value is said to be relative. For a single given sample, the relative atomic mass of a given element is the weighted arithmetic mean of the masses of the individual atoms (including their isotopes) that are present in the sample.
|
|
The book combined thermodynamics theory with engineering heat transfer and fluid mechanics, and introduced entropy generation minimization as a method of optimization. In 1996 the ASME awarded him the Worcester Reed Warner Medal for "originality, challenges to orthodoxy, and impact on thermodynamics and heat transfer, which were made through his first three books". In 1989 Bejan was appointed the J. A. Jones Distinguished Professor of Mechanical Engineering. In 1988 and 1989, his peers named two dimensionless groups Bejan number (Be), in two different fields: for the pressure difference group, in heat transfer by forced convection, and for the dimensionless ratio of fluid friction irreversibility divided by heat transfer irreversibility, in thermodynamics.
|
|
Then atoms would be bigger (in one dimension) by 2, each of us would be taller by 2, and so would our metre sticks be taller (and wider and thicker) by a factor of 2. Our perception of distance and lengths relative to the Planck length is, by axiom, an unchanging dimensionless constant. Our clocks would tick slower by a factor of 4 (from the point of view of this unaffected observer on the outside) because the Planck time has increased by 4 but we would not know the difference (our perception of durations of time relative to the Planck time is, by axiom, an unchanging dimensionless constant).
|
|
The LRL vector has been re-discovered several times and is also equivalent to the dimensionless eccentricity vector of celestial mechanics. Various generalizations of the LRL vector have been defined, which incorporate the effects of special relativity, electromagnetic fields and even different types of central forces.
|
|
The Stanton number, St, is a dimensionless number that measures the ratio of heat transferred into a fluid to the thermal capacity of fluid. The Stanton number is named after Thomas Stanton (engineer) (1865–1931). It is used to characterize heat transfer in forced convection flows.
|
|
A point particle (ideal particleH. C. Ohanian, J. T. Markert (2007), p. 3. or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up space.
|
|
John Ronald Womersley (20 June 1907 - 7 March 1958) was a British mathematician and computer scientist who made important contributions to computer development, and hemodynamics. Nowadays he is principally remembered for his contribution to blood flow, fluid dynamics and the eponymous Womersley number, a dimensionless parameter characterising unsteady flow.
|
|
There are different scales, denoted by a single letter, that use different loads or indenters. The result is a dimensionless number noted as HRA, HRB, HRC, etc., where the last letter is the respective Rockwell scale (see below). When testing metals, indentation hardness correlates linearly with tensile strength.
|
|
The unit was formerly an SI supplementary unit (before that category was abolished in 1995) and the radian is now considered an SI derived unit. The radian is defined in the SI as being a dimensionless value, and its symbol is accordingly often omitted, especially in mathematical writing.
|
|
Just like the radian, the milliradian is dimensionless, but unlike the radian where the same unit must be used for radius and arc length, the milliradian needs to have a ratio between the units where the subtension is a thousandth of the radius when using the simplified formula.
|
|
Higher combined Ec/Io, lower traffic channel Ec/Io is required and more BTS power is conserved. Ec/Io is a notation used to represent a dimensionless ratio of the average power of a channel, typically the pilot channel, to the total signal power. It is expressed in dB.
|
|
Here, the emitting power denotes a dimensioned quantity, the total radiation emitted by a body labeled by index at temperature . The total absorption ratio of that body is dimensionless, the ratio of absorbed to incident radiation in the cavity at temperature . (In contrast with Balfour Stewart's, Kirchhoff's definition of his absorption ratio did not refer in particular to a lamp-black surface as the source of the incident radiation.) Thus the ratio of emitting power to absorptivity is a dimensioned quantity, with the dimensions of emitting power, because is dimensionless. Also here the wavelength-specific emitting power of the body at temperature is denoted by and the wavelength-specific absorption ratio by .
|
|
In engineering, applied mathematics, and physics, the Buckingham theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically meaningful equation involving a certain number n of physical variables, then the original equation can be rewritten in terms of a set of p = n − k dimensionless parameters 1, 2, ..., p constructed from the original variables. (Here k is the number of physical dimensions involved; it is obtained as the rank of a particular matrix.) The theorem provides a method for computing sets of dimensionless parameters from the given variables, or nondimensionalization, even if the form of the equation is still unknown.
|
|
In the point of view of engineering, calculating skin friction is useful in estimating not only total frictional drag exerted on an object but also convectional heat transfer rate on its surface. This relationship is well developed in the concept of Reynolds analogy, which links two dimensionless parameters: skin friction coefficient (Cf), which is a dimensionless frictional stress, and Nusselt number (Nu), which indicates the magnitude of convectional heat transfer. Turbine blades, for example, require the analysis of heat transfer in their design process since they are imposed in high temperature gas, which can damage them with the heat. Here, engineers calculate skin friction on the surface of turbine blades to predict heat transfer occurred through the surface.
|
|
In the design of fluid bearings, the Sommerfeld number (S) is a dimensionless quantity used extensively in hydrodynamic lubrication analysis. The Sommerfeld number is very important in lubrication analysis because it contains all the variables normally specified by the designer. The Sommerfeld number is named after Arnold Sommerfeld (1868–1951).
|
|
The dimensionless quantities often represent the degree of deviation from an ideal shape, such as a circle, sphere or equilateral polyhedron.L. Wojnar & K.J. Kurzydłowski, et al., Practical Guide to Image Analysis, ASM International, 2000, p 157-160, . Shape factors are often normalized, that is, the value ranges from zero to one.
|
|
Pycnoclines become unstable when their Richardson number drops below 0.25. The Richardson number is a dimensionless value expressing the ratio of potential to kinetic energy. This ratio drops below 0.25 when the shear rate exceeds stratification. This can produce Kelvin- Helmholtz instability, resulting in a turbulence which leads to mixing.
|
|
The heat transfer coefficient is often calculated from the Nusselt number (a dimensionless number). There are also online calculators available specifically for Heat-transfer fluid applications. Experimental assessment of the heat transfer coefficient poses some challenges especially when small fluxes are to be measured (e.g. < 0.2 \rm W/cm^2 ).
|
|
Optical magnification is the ratio between the apparent size of an object (or its size in an image) and its true size, and thus it is a dimensionless number. Optical magnification is sometimes referred to as "power" (for example "10× power"), although this can lead to confusion with optical power.
|
|
Characteristic numbers are dimensionless numbers used in fluid dynamics to describe a character of the flow. To compare a real situation (e.g. an aircraft) with a small-scale model it is necessary to keep the important characteristic numbers the same. Names of these numbers were standardized in ISO 31, part 12.
|
|
Luminous efficacy can be normalized by the maximum possible luminous efficacy to a dimensionless quantity called luminous efficiency. The distinction between efficacy and efficiency is not always carefully maintained in published sources, so it is not uncommon to see "efficiencies" expressed in lumens per watt, or "efficacies" expressed as a percentage.
|
|
The Optical Unit is a dimensionless units of length used in optical microscopy. Because every diffraction limited system have their resolution proportional to wavelength / NA, it is convenient for comparison to use this unit. There are actually 2 units, one "axial" (along the optical axis of the objective) and one "radial".
|
|
The dBm is also dimensionless but since it compares to a fixed reference value the dBm rating is an absolute one. In audio and telephony, dBm is typically referenced relative to a 600-ohm impedance, while in radio-frequency work dBm is typically referenced relative to a 50-ohm impedance.
|
|
Drag and lift coefficients for NACA 633618 airfoil. Full curves are lift, dashed drag; red curves have = 9·106, blue 3·106. Drag polar for the NACA 633618 airfoil, colour-coded as opposite plot. The significant aerodynamic properties of aircraft wings are summarised by two dimensionless quantities, the lift and drag coefficients and .
|
|
Tegmark et al. have recently considered these objections and proposed a simplified anthropic scenario for axion dark matter in which they argue that the first two of these problems do not apply.M. Tegmark, A. Aguirre, M. Rees and F. Wilczek, "Dimensionless constants, cosmology and other dark matters", . F. Wilczek, "Enlightenment, knowledge, ignorance, temptation", .
|
|
While one may informally say "the molar mass of an element M is the same as its atomic weight A", the atomic weight (relative atomic mass) A is a dimensionless quantity, whereas the molar mass M has the units of mass per mole. Formally, M is A times the molar mass constant Mu.
|
|
In statistics, the analogous process is usually dividing a difference (a distance) by a scale factor (a measure of statistical dispersion), which yields a dimensionless number, which is called normalization. Most often, this is dividing errors or residuals by the standard deviation or sample standard deviation, respectively, yielding standard scores and studentized residuals.
|
|
In human gait, to travel a particular distance, chemical energy must be expended by the body. This relationship can be expressed by the dimensionless term, cost of transport (COT).,Ralston, H. J. (1958). Energy-speed relation and optimal speed during level walking. Internationale Zeitschrift für Angewandte Physiologie Einschliesslich Arbeitsphysiologie, 17(4), 277-283.
|
|
George Gamow argued in his book Mr Tompkins in Wonderland that a sufficient change in a dimensionful physical constant, such as the speed of light in a vacuum, would result in obvious perceptible changes. But this idea is challenged: Referring to Duff's "Comment on time-variation of fundamental constants" and Duff, Okun, and Veneziano's paper "Trialogue on the number of fundamental constants", particularly the section entitled "The operationally indistinguishable world of Mr. Tompkins", if all physical quantities (masses and other properties of particles) were expressed in terms of Planck units, those quantities would be dimensionless numbers (mass divided by the Planck mass, length divided by the Planck length, etc.) and the only quantities that we ultimately measure in physical experiments or in our perception of reality are dimensionless numbers. When one commonly measures a length with a ruler or tape-measure, that person is actually counting tick marks on a given standard or is measuring the length relative to that given standard, which is a dimensionless value. It is no different for physical experiments, as all physical quantities are measured relative to some other like-dimensioned quantity.
|
|
This followed a naming contest which attracted 6,000 entries from fifty states and sixty-one countries. The Chandra X-ray Observatory was launched and deployed by Space Shuttle Columbia on 23 July 1999. The Chandrasekhar number, an important dimensionless number of magnetohydrodynamics, is named after him. The asteroid 1958 Chandra is also named after Chandrasekhar.
|
|
The Chandrasekhar number is a dimensionless quantity used in magnetic convection to represent ratio of the Lorentz force to the viscosity. It is named after the Indian astrophysicist Subrahmanyan Chandrasekhar. The number's main function is as a measure of the magnetic field, being proportional to the square of a characteristic magnetic field in a system.
|
|
In planetary sciences, the moment of inertia factor or normalized polar moment of inertia is a dimensionless quantity that characterizes the radial distribution of mass inside a planet or satellite. Since a moment of inertia must have dimensions of mass times length squared, the moment of inertia factor is the coefficient that multiplies these.
|
|
Johansson proposes that there are logical flaws in the application of quantity calculus, and that the so- called dimensionless quantities should be understood as "unitless quantities". How to use quantity calculus for unit conversion and keeping track of units in algebraic manipulations is explained in the handbook on Quantities, Units and Symbols in Physical Chemistry.
|
|
The Stuart number (N), also known as magnetic interaction parameter, is a dimensionless number of fluids, i.e. gases or liquids. It is defined as the ratio of electromagnetic to inertial forces, which gives an estimate of the relative importance of a magnetic field on a flow. The Stuart number is relevant for flows of conducting fluids, e.g.
|
|
The Hatta number (Ha) was developed by Shirôji Hatta, who taught at Tohoku University.S. Hatta, Technological Reports of Tôhoku University, 10, 613-622 (1932). It is a dimensionless parameter that compares the rate of reaction in a liquid film to the rate of diffusion through the film.R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, 2nd ed.
|
|
Remove these powers, and the conception of matter vanishes into space—conceive repulsion only, and you have the same result. For infinite repulsion, uncounteracted and alone, is tantamount to infinite, dimensionless diffusion, and this again to infinite weakness; viz., to space. Conceive attraction alone, and as an infinite contraction, its product amounts to the absolute point, viz.
|
|
As a simplified example, if a beamline runs for 8 hours (28 800 seconds) at an instantaneous luminosity of , then it will gather data totaling an integrated luminosity of = = during this period. If this is multiplied by the cross-section, then a dimensionless number is obtained which would be simply the number of expected scattering events.
|
|
The expectation value of in the ground state (the vacuum expectation value or VEV) is then , where . The measured value of this parameter is approximately . It has units of mass, and is the only free parameter of the Standard Model that is not a dimensionless number. The Higgs mechanism is a type of superconductivity which occurs in the vacuum.
|
|
Physics often uses dimensionless quantities to simplify the characterization of systems with multiple interacting physical phenomena. These may be found by applying the Buckingham theorem or otherwise may emerge from making partial differential equations unitless by the process of nondimensionalization. Engineering, economics, and other fields often extend these ideas in design and analysis of the relevant systems.
|
|
But dimensional analysis can be a powerful aid in understanding problems like this, and is usually the very first tool to be applied to complex problems where the underlying equations and constraints are poorly understood. In such cases, the answer may depend on a dimensionless number such as the Reynolds number, which may be interpreted by dimensional analysis.
|
|
The Cohesion number (Coh) is a useful dimensionless number in particle technology by which the cohesivity of different powders can be compared. This is especially useful in DEM simulations (Discrete Element Method) of granular materials where scaling of the size and stiffness of the particles are inevitable due to the computationally demanding nature of the DEM modelling.
|
|
David McGoveran (born 1952) is an American computer scientist and physicist, software industry analyst, and inventor. In computer science, he is recognized as one of the pioneers of relational database theory. In the field of physics, his most notable work is in discrete and bit-string physics, in which he derived fundamental dimensionless constants from first principles.
|
|
For example, reliability of a scheduled aircraft flight can be specified as a dimensionless probability or a percentage, as often used in system safety engineering. A special case of mission success is the single-shot device or system. These are devices or systems that remain relatively dormant and only operate once. Examples include automobile airbags, thermal batteries and missiles.
|
|
In physics, a gravitational coupling constant is a constant characterizing the gravitational attraction between a given pair of elementary particles. The electron mass is typically used, and the associated constant typically denoted . It is a dimensionless quantity, with the result that its numerical value does not vary with the choice of units of measurement, only with the choice of particle.
|
|
The dynamic of fluid parcels is described with the help of Newton's second law. An accelerating parcel of fluid is subject to inertial effects. The Reynolds number is a dimensionless quantity which characterises the magnitude of inertial effects compared to the magnitude of viscous effects. A low Reynolds number (Re ≪ 1) indicates that viscous forces are very strong compared to inertial forces.
|
|
For a specific propeller geometry, Kt and Kq are often given as a function of the advance number J. It is a dimensionless number indicating some speed. It has all the components of how fast the rpm should be. These co-efficients are experimentally determined by so-called open water tests, usually performed in a cavitation tunnel or a towing tank.
|
|
The two major tidal forces acting on Mars are the solar tide and Phobos tide. Love number k2 is an important proportional dimensionless constant relating the tidal field acting to the body with the multipolar moment resulting from the mass distribution of the body. Usually k2 can tell quadrupolar deformation. Finding k2 is helpful in understanding the interior structure on Mars.
|
|
An unmoved mover is assumed for each sphere, including a "prime mover" for the sphere of fixed stars. The unmoved movers do not push the spheres (nor could they, being immaterial and dimensionless) but are the final cause of the spheres' motion, i.e. they explain it in a way that's similar to the explanation "the soul is moved by beauty".
|
|
In radio frequency (RF) practice this is often measured in a dimensionless ratio known as voltage standing wave ratio (VSWR) with a VSWR bridge. The ratio of energy bounced back depends on the impedance mismatch. Mathematically, it is defined using the reflection coefficient. Because the principles are the same, this concept is perhaps easiest to understand when considering an optical fiber.
|
|
In the fantasy novels of Feist, a Riftwar is war between two worlds that are connected by some sort of dimensionless gap (rift). In Feist's invented history there are several riftwars. The first Riftwar between Midkemia and Kelewan is described in the trilogy The Riftwar Saga. This Saga is a continuation of Feist's preceding works and so far suggests an upcoming, fourth riftwar.
|
|
The constants α and x can be estimated through iteration from a given species data set using the values S and N.Magurran, A. E. 2004. Measuring biological diversity. Blackwell Scientific, Oxford. Fisher's dimensionless α is often used as a measure of biodiversity, and indeed has recently been found to represent the fundamental biodiversity parameter θ from neutral theory (see below).
|
|
In measuring unsaturation in fatty acids, the traditional method is the iodine number. Iodine adds stoichiometrically to double bonds, so their amount is reported in grams of iodine spent per 100 grams of oil. The standard unit is a dimensionless stoichiometry ratio of moles double bonds to moles fatty acid. A similar quantity, bromine number, is used in gasoline analysis.
|
|
Multiplying any quantity (physical quantity or not) by the dimensionless 1 does not change that quantity. Once this and the conversion factor for seconds per hour have been multiplied by the original fraction to cancel out the units mile and hour, 10 miles per hour converts to 4.4704 meters per second. As a more complex example, the concentration of nitrogen oxides (i.e.
|
|
The respiratory quotient (or RQ or respiratory coefficient), is a dimensionless number used in calculations of basal metabolic rate (BMR) when estimated from carbon dioxide production. It is calculated from the ratio of carbon dioxide produced by the body to oxygen consumed by the body. Such measurements, like measurements of oxygen uptake, are forms of indirect calorimetry. It is measured using a respirometer.
|
|
In spectroscopy, oscillator strength is a dimensionless quantity that expresses the probability of absorption or emission of electromagnetic radiation in transitions between energy levels of an atom or molecule. The oscillator strength can be thought of as the ratio between the quantum mechanical transition rate and the classical absorption/emission rate of a single electron oscillator with the same frequency as the transition.
|
|
So their RpW ratio = 38.214 ÷ 26.7 = 1.431. The RpW ratio for their opponent (New Zealand) is the inverse of this: 26.7 ÷ 38.214 = 0.699. So if two teams have played only each other, their two RpW ratio figures are reciprocals. As the units are the same either side of the division (runs/wickets), they cancel out, so RpW ratio is a dimensionless quantity.
|
|
In fluid dynamics, the Nusselt number (Nu) is the ratio of convective to conductive heat transfer at a boundary in a fluid. Convection includes both advection (fluid motion) and diffusion (conduction). The conductive component is measured under the same conditions as the convective but for a hypothetically motionless fluid. It is a dimensionless number, closely related to the fluid's Rayleigh number.
|
|
The pressure coefficient is a dimensionless number which describes the relative pressures throughout a flow field in fluid dynamics. The pressure coefficient is used in aerodynamics and hydrodynamics. Every point in a fluid flow field has its own unique pressure coefficient, C_p. In many situations in aerodynamics and hydrodynamics, the pressure coefficient at a point near a body is independent of body size.
|
|
Constrictivity is a dimensionless parameter used to describe transport processes (often molecular diffusion) in porous media. Constrictivity is viewed to depend on the ratio of the diameter of the diffusing particle to the pore diameter. The value of constrictivity is always less than 1. The constrictivity is defined not for a single pore, but as the parameter of the entire pore space considered.
|
|
This would result in a reduced speed exponent. This factor is not likely to explain the large reduction in exponent however. The blade geometry was highly variable in the tests, so it is likely that the negative dependence of the dimensionless force on Reynolds number is the major factor. This whistle has two features that separate it from the other whistles described here.
|
|
Line of impact – It is the line along which e is defined or in absence of tangential reaction force between colliding surfaces, force of impact is shared along this line between bodies. During physical contact between bodies during impact its line along common normal to pair of surfaces in contact of colliding bodies. Hence e is defined as a dimensionless one- dimensional parameter.
|
|
The Bulk Richardson Number (BRN) is a dimensionless number relating vertical stability and vertical wind shear (generally, stability divided by shear). It represents the ratio of thermally-produced turbulence and turbulence generated by vertical shear. Practically, its value determines whether convection is free or forced. High values indicate unstable and/or weakly sheared environments; low values indicate weak instability and/or strong vertical shear.
|
|
Examples of gluon coupling Particles which interact with each other are said to be coupled. This interaction is caused by one of the fundamental forces, whose strengths are usually given by a dimensionless coupling constant. In quantum electrodynamics, this value is known as the fine-structure constant α, approximately equal to 1/137. For quantum chromodynamics, the constant changes with respect to the distance between the particles.
|
|
CO;2 Nurser, A. J. G., & Bacon, S., 2014, The Rossby radius in the Arctic Ocean, Ocean Sci., 10, 967-975, doi: 10.5194/os-10-967-2014 The size of ocean eddies varies similarly; in low latitude regions, near the equator, eddies are much larger than in high latitude regions. The associated dimensionless parameter is the Rossby number. Both are named in honor of Carl-Gustav Rossby.
|
|
Figure 2. Probability density functions of the molecular speed for four noble gases at a temperature of 298.15 K (25 °C). The four gases are helium (4He), neon (20Ne), argon (40Ar) and xenon (132Xe); the superscripts indicate their mass numbers. These probability density functions have dimensions of probability times inverse speed; since probability is dimensionless, they can be expressed in units of seconds per meter.
|
|
From simple plucking with the thumb and index finger and saoxian (sweeping one's fingers across all strings with gusto) to yaozhi (tilting the instrument and using the middle finger to continuously cut across the strings) and lunzou (by plucking with all five fingers, one after another in a wavelike motion), the playing techniques of the pipa are visibly dimensionless. Its tuning is A2-D3-E3-A3.
|
|
The damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance. Many systems exhibit oscillatory behavior when they are disturbed from their position of static equilibrium. A mass suspended from a spring, for example, might, if pulled and released, bounce up and down. On each bounce, the system tends to return to its equilibrium position, but overshoots it.
|
|
The Deborah number (De) is a dimensionless number, often used in rheology to characterize the fluidity of materials under specific flow conditions. It quantifies the observation that given enough time even a solid-like material might flow, or a fluid-like material can act solid when it is deformed rapidly enough. Materials that have low relaxation times flow easily and as such show relatively rapid stress decay.
|
|
The units must be consistent, e.g. f may be in metres, \omega in radians per second, and g in metres per second-squared. If we write F for the numerical value of the focal length in metres, and S for the numerical value of the rotation speed in revolutions per minute (RPM),Thus F and S are dimensionless numbers. 30 RPM = \pi radians per second.
|
|
With this change, the international prototype of the kilogram is being retired as the last physical object used in the definition of any SI unit. Tests on the immutability of physical constants look at dimensionless quantities, i.e. ratios between quantities of like dimensions, in order to escape this problem. Changes in physical constants are not meaningful if they result in an observationally indistinguishable universe.
|
|
The erlang, named after A. K. Erlang, as a dimensionless unit is used in telephony as a statistical measure of the offered intensity of telecommunications traffic on a group of resources. Traffic of one erlang refers to a single resource being in continuous use, or two channels being at fifty percent use, and so on, pro rata. Much telecommunications management and forecasting software uses this.
|
|
Numerous dimensionless numbers, mostly ratios, were coined in the early 1900s, particularly in the areas of fluid mechanics and heat transfer. Measuring ratios in the (derived) unit dB (decibel) finds widespread use nowadays. In the early 2000s, the International Committee for Weights and Measures discussed naming the unit of 1 as the "uno", but the idea of just introducing a new SI name for 1 was dropped.
|
|
This threshold is governed by the Reynolds Number, a dimensionless number in fluid dynamics that is directly proportional to fluid velocity. Eruptions will be effusive if the magma has a low ascent velocity. At higher magma ascent rates, the fragmentation within the magma passes a threshold and results in explosive eruptions. Silicic magma also exhibits this transition between effusive and explosive eruptions, but the fragmentation mechanism differs.
|
|
In North America, the hardness of water is often measured in grains per US gallon (gpg) of calcium carbonate equivalents. Otherwise, water hardness is measured in the dimensionless unit of parts per million (ppm), numerically equivalent to density measured in mg/L. One grain per US gallon is approximately . Soft water contains 1–4 gpg of calcium carbonate equivalents, while hard water contains 11–20 gpg.
|
|
These features are commonly associated with many active galaxies. The axial ratio of the elliptical galaxy is 1.4, meaning it is about 1.4 times large along the primary axis than along the perpendicular axis. At the nucleus of this galaxy is a supermassive black hole with an estimated solar masses. The dimensionless ratio of the black hole spin to the black hole mass-energy j is .
|
|
In that same year, he fell ill with pancreatic cancer. When his last assistant, Charles Enz, visited him at the Rotkreuz hospital in Zurich, Pauli asked him: "Did you see the room number?" It was number 137. Throughout his life, Pauli had been preoccupied with the question of why the fine structure constant, a dimensionless fundamental constant, has a value nearly equal to 1/137.
|
|
The nodes of the signal-flow graph will include both voltages and currents. The branch gains will include impedances and admittances. :4. Convert all nodes of the signal-flow graph to voltages and all impedances to dimensionless transmittances. This is accomplished by dividing all impedance elements by R, an arbitrary resistance and multiplying all admittance elements by R. This scaling does not change the frequency response. :5.
|
|
These photographs resemble prehistoric cave-paintings: the black, dimensionless spaces on glass. Fossil- like facial forms and dismembered body parts coexist uncomfortably with vaporous, ghost-like shadows. In an accompanying film animation, Roger Ballen's Theatre of the Apparitions (2016), Emma Calder and Ged Haney created an animated theatre of the book's dismembered people, beasts and ghosts, dance, tumble, make love and tear themselves apart a nightmarish subconscious world.
|
|
Tahmasebi, H.A., Kharrat, R., Soltanieh, M.: Dimensionless correlation for prediction of permeability reduction rate due to calcium sulfate scale deposition in carbonate grain packed column, J. of the Taiwan Institute of Chemical Engineers, Vol. 4 No. 3 May 2010, pp268–278. 53\. Ghazanfari M. H., Kharrat R., Rashtchian D. Vossoughi S., Statistical Model for Dispersion in a 2D Glass Micromodel, SPE Journal, Vol. 15, No. 2, June 2010, pp301–312. 54\.
|
|
As noted above, the results of PCA depend on the scaling of the variables. This can be cured by scaling each feature by its standard deviation, so that one ends up with dimensionless features with unital variance.Leznik, M; Tofallis, C. 2005 Estimating Invariant Principal Components Using Diagonal Regression. The applicability of PCA as described above is limited by certain (tacit) assumptionsJonathon Shlens, A Tutorial on Principal Component Analysis.
|
|
Hysteresivity derives from “hysteresis”, meaning “lag”. It is the tendency to react slowly to an outside force, or to not return completely to its original state. Whereas the area within a hysteresis loop represents energy dissipated to heat and is an extensive quantity with units of energy, the hysteresivity represents the fraction of the elastic energy that is lost to heat, and is an intensive property that is dimensionless.
|
|
All physical quantities are identified with geometric quantities such as areas, lengths, dimensionless numbers, path curvatures, or sectional curvatures. Many equations in relativistic physics appear simpler when expressed in geometric units, because all occurrences of G and of c drop out. For example, the Schwarzschild radius of a nonrotating uncharged black hole with mass m becomes . For this reason, many books and papers on relativistic physics use geometric units.
|
|
One feature of Boussinesq flows is that they look the same when viewed upside-down, provided that the identities of the fluids are reversed. The Boussinesq approximation is inaccurate when the dimensionless density difference is of order unity. For example, consider an open window in a warm room. The warm air inside is less dense than the cold air outside, which flows into the room and down towards the floor.
|
|
The point at which this happened was the transition point from laminar to turbulent flow. Reynolds identified the governing parameter for the onset of this effect, which was a dimensionless constant later called the Reynolds number. Reynolds found that the transition occurred between Re = 2000 and 13000, depending on the smoothness of the entry conditions. When extreme care is taken, the transition can even happen with Re as high as 40000.
|
|
Large scale means that the scale is much larger than the facet size. This equation is double sin-Gordon equation, which in normalized units reads where is a dimensionless constant resulting from averaging over tiny facets. The detailed mathematical procedure of averaging is similar to the one done for a parametrically driven pendulum, and can be extended to time-dependent phenomena. In essence, () described extended φ Josephson junction.
|
|
If there is only one zero, then gT2/L = C. It requires more physical insight or an experiment to show that there is indeed only one zero and that the constant is in fact given by C = 4π2. For large oscillations of a pendulum, the analysis is complicated by an additional dimensionless parameter, the maximum swing angle. The above analysis is a good approximation as the angle approaches zero.
|
|
For a pure substance the density has the same numerical value as its mass concentration. Different materials usually have different densities, and density may be relevant to buoyancy, purity and packaging. Osmium and iridium are the densest known elements at standard conditions for temperature and pressure. To simplify comparisons of density across different systems of units, it is sometimes replaced by the dimensionless quantity "relative density" or "specific gravity", i.e.
|
|
The erlang (symbol EHow Many? A Dictionary of Units of Measurement) is a dimensionless unit that is used in telephony as a measure of offered load or carried load on service-providing elements such as telephone circuits or telephone switching equipment. A single cord circuit has the capacity to be used for 60 minutes in one hour. Full utilization of that capacity, 60 minutes of traffic, constitutes 1 erlang.
|
|
After the war Bagnold continued to work in the field of the geological science, and he published academic papers into his nineties. He made significant contributions to the understanding of desert terrain such as sand dunes, ripples and sheets. He developed the dimensionless "Bagnold number" and "Bagnold formula" for characterising sand flow. He gave a constitutive relation for a suspension of neutrally buoyant particles in a Newtonian fluid.
|
|
For this reason turbulence is commonly realized in low viscosity fluids. In general terms, in turbulent flow, unsteady vortices appear of many sizes which interact with each other, consequently drag due to friction effects increases. This increases the energy needed to pump fluid through a pipe. The onset of turbulence can be predicted by the dimensionless Reynolds number, the ratio of kinetic energy to viscous damping in a fluid flow.
|
|
Smith, Warren Modern Lens Design 2005 McGraw-Hill. It is a dimensionless number that is a quantitative measure of lens speed; increasing the f-number is referred to as stopping down. The f-number is commonly indicated using a hooked f with the format N, where N is the f-number. The f-number is the reciprocal of the relative aperture (the aperture diameter divided by focal length).
|
|
The atomic mass (mr) of an isotope (nuclide) is determined mainly by its mass number (i.e. number of nucleons in its nucleus). Small corrections are due to the binding energy of the nucleus (see mass defect), the slight difference in mass between proton and neutron, and the mass of the electrons associated with the atom, the latter because the electron:nucleon ratio differs among isotopes. The mass number is a dimensionless quantity.
|
|
This curve has no more than a historical value nowadays, although its simplicity is still attractive. Among the drawbacks of this curve are that it does not take the water depth into account and more importantly, that it does not show that sedimentation is caused by flow velocity deceleration and erosion is caused by flow acceleration. The dimensionless Shields diagram is now unanimously accepted for initiation of sediment motion in rivers.
|
|
This dimensionless quantity can also be rephrased as the "actual path length" divided by the "shortest path length" of a curve. The value ranges from 1 (case of straight line) to infinity (case of a closed loop, where the shortest path length is zero or for an infinitely-long actual pathLeopold, Luna B., Wolman, M.G., and Miller, J.P., 1964, Fluvial Processes in Geomorphology, San Francisco, W.H. Freeman and Co., 522p.).
|
|
The logarithmic scale can compactly represent the relationship among variously sized numbers. This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantity and probabilities. Each number is given a name in the short scale, which is used in English- speaking countries, as well as a name in the long scale, which is used in some of the countries that do not have English as their national language.
|
|
As we already saw above, the nematic liquid crystals are composed of rod-like molecules with the long axes of neighboring molecules aligned approximately to one another. To describe this anisotropic structure, a dimensionless unit vector n called the director, is introduced to represent the direction of preferred orientation of molecules in the neighborhood of any point. Because there is no physical polarity along the director axis, n and -n are fully equivalent.
|
|
Also, other changes of the coordinate system may affect the formula for computing the scalar (for example, the Euclidean formula for distance in terms of coordinates relies on the basis being orthonormal), but not the scalar itself. In this sense, physical distance deviates from the definition of metric in not being just a real number; however it satisfies all other properties. The same applies for other physical quantities which are not dimensionless.
|
|
That is one aspect of determining the size of the droplet is the velocity of liquid in the channel (v=Rω). So, we have four dimensionless terms derived from the above properties which determine the performance of atomization. 1\. Liquid-gas density ratio r = [ρL / ρG] where ρL and ρG are densities of liquid and gas respectively 2\. Viscosity ratio m = [µL / µG] where, µL and µG are viscosities of liquid and gas respectively 3\.
|
|
Ductile iron pipe is sized according to a dimensionless term known as the Pipe Size or Nominal Diameter (known by its French abbreviation, DN). This is roughly equivalent to the pipe's internal diameter in inches or millimeters. However, it is the external diameter of the pipe that is kept constant between changes in wall thickness, in order to maintain compatibility in joints and fittings. Consequently, the internal diameter varies, sometimes significantly, from its nominal size.
|
|
Illustration showing the face of a Machmeter reading a Mach number of 0.83 A Machmeter is an aircraft pitot-static system flight instrument that shows the ratio of the true airspeed to the speed of sound, a dimensionless quantity called Mach number. This is shown on a Machmeter as a decimal fraction. An aircraft flying at the speed of sound is flying at a Mach number of one, expressed as Mach 1\.
|
|
Thus, for all practical purposes, this is the same as the simple dimensionless BAC measured as a percent. The per mille (promille) measurement, which is equal to ten times the percentage value, is used in Denmark, Germany, Finland, Norway and Sweden. Depending on the jurisdiction, BAC may be measured by police using three methods: blood, breath, or urine. For law enforcement purposes, breath is the preferred method, since results are available almost instantaneously.
|
|
Attempts to find a mathematical basis for this dimensionless constant have continued up to the present time. However, no numerological explanation has ever been accepted by the physics community. In the early 21st century, multiple physicists, including Stephen Hawking in his book A Brief History of Time, began exploring the idea of a multiverse, and the fine-structure constant was one of several universal constants that suggested the idea of a fine-tuned universe.
|
|
In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number. There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is usually required to compute the area.
|
|
The large majority of an atom's mass comes from the protons and neutrons that make it up. The total number of these particles (called "nucleons") in a given atom is called the mass number. It is a positive integer and dimensionless (instead of having dimension of mass), because it expresses a count. An example of use of a mass number is "carbon-12," which has 12 nucleons (six protons and six neutrons).
|
|
Canopy conductance, commonly denoted g_c, is a dimensionless quantity characterizing radiation distribution in tree canopy. By definition, it is calculated as a ratio of daily water use to daily mean vapor pressure deficit (VPD). Canopy conductance can be also experimentally obtained by measuring sap flow and environmental variables. Stomatal conductance may be used as a reference value to validate the data, by summing the total stomatal conductance g_s of all leaf classes within the canopy.
|
|
Tests on the immutability of physical constants look at dimensionless quantities, i.e. ratios between quantities of like dimensions, in order to escape this problem. Changes in physical constants are not meaningful if they result in an observationally indistinguishable universe. For example, a "change" in the speed of light c would be meaningless if accompanied by a corresponding "change" in the elementary charge e so that the ratio e2:c (the fine-structure constant) remained unchanged.
|
|
The performance of vapor compression refrigeration cycles is limited by thermodynamics. These air conditioning and heat pump devices move heat rather than convert it from one form to another, so thermal efficiencies do not appropriately describe the performance of these devices. The Coefficient-of- Performance (COP) measures performance, but this dimensionless measure has not been adopted. Instead, the Energy Efficiency Ratio (EER) has traditionally been used to characterize the performance of many HVAC systems.
|
|
The coefficient of variation is useful because the standard deviation of data must always be understood in the context of the mean of the data. In contrast, the actual value of the CV is independent of the unit in which the measurement has been taken, so it is a dimensionless number. For comparison between data sets with different units or widely different means, one should use the coefficient of variation instead of the standard deviation.
|
|
In fluid mechanics, non-dimensionalization of the Navier–Stokes equations is the conversion of the Navier–Stokes equation to a nondimensional form. This technique can ease the analysis of the problem at hand, and reduce the number of free parameters. Small or large sizes of certain dimensionless parameters indicate the importance of certain terms in the equations for the studied flow. This may provide possibilities to neglect terms in certain (areas of the) considered flow.
|
|
He found the number to be reasonably constant over given a Reynolds number range. This number permits relationships to be developed between the different sizes and speeds. Now the Strouhal number can be derived directly from the dimensionless form of the mass continuity equation. This equation may be referred to as a fluid mechanical Strouhal number in comparison with the second version, which may be referred to as the acoustical Strouhal number.
|
|
The Knudsen number (Kn) is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale. This length scale could be, for example, the radius of a body in a fluid. The number is named after Danish physicist Martin Knudsen (1871–1949). The Knudsen number helps determine whether statistical mechanics or the continuum mechanics formulation of fluid dynamics should be used to model a situation.
|
|
The Dean number (De) is a dimensionless group in fluid mechanics, which occurs in the study of flow in curved pipes and channels. It is named after the British scientist W. R. Dean, who was the first to provide a theoretical solution of the fluid motion through curved pipes for laminar flow by using a perturbation procedure from a Poiseuille flow in a straight pipe to a flow in a pipe with very small curvature.
|
|
This is a list of voids in astronomy. Voids are particularly galaxy-poor regions of space between filaments, making up the large-scale structure of the universe. Some voids are known as supervoids. A map of galaxy voids In the tables, z is the cosmological redshift, c the speed of light, and h the dimensionless Hubble parameter, which has a value of approximately 0.7 (the Hubble constant H0 = h × 100 km s−1 Mpc−1).
|
|
Fluorescein aqueous solutions, diluted from 1 to 10,000 parts-per-million 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 (ppm, ), parts-per-billion (ppb, ), parts- per-trillion (ppt, ) and parts-per-quadrillion (ppq, ).
|
|
The majority of Feist's works are part of The Riftwar Universe, and feature the worlds of Midkemia and Kelewan. Human magicians and other creatures on the two planets are able to create rifts through dimensionless space that can connect planets in different solar systems. The novels and short stories of The Riftwar Universe record the adventures of various people on these worlds. Midkemia was originally created as an alternative to the Dungeons and Dragons (D&D;) role-playing game.
|
|
This is a dimensionless quantity (i.e., a pure number, without units) equal to the molar mass divided by the molar mass constant.The technical definition is that the relative molar mass is the molar mass measured on a scale where the molar mass of unbound carbon 12 atoms, at rest and in their electronic ground state, is 12. The simpler definition given here is equivalent to the full definition because of the way the molar mass constant is itself defined.
|
|
For instance, the flow around a circular cylinder generates a Kármán vortex street: vortices being shed in an alternating fashion from the cylinder's sides. The oscillatory nature of the flow produces a fluctuating lift force on the cylinder, even though the net (mean) force is negligible. The lift force frequency is characterised by the dimensionless Strouhal number, which depends on the Reynolds number of the flow. For a flexible structure, this oscillatory lift force may induce vortex-induced vibrations.
|
|
Usually, as a plume moves away from its source, it widens because of entrainment of the surrounding fluid at its edges. Plume shapes can be influenced by flow in the ambient fluid (for example, if local wind blowing in the same direction as the plume results in a co-flowing jet). This usually causes a plume which has initially been 'buoyancy-dominated' to become 'momentum-dominated' (this transition is usually predicted by a dimensionless number called the Richardson number).
|
|
Classical or topological ordering systems are assigned a dimensionless numerical order of "one", starting at the mouth of a stream, which is its lowest elevation point. The vector order then increases as it traces upstream and converges with other smaller streams, resulting in a correlation of higher-order numbers to more highly-elevated headwaters. Horton proposed to establish a reversal of that order. Horton's 1947 research report established a stream ordering method based on vector geometry.
|
|
Ratios may be unitless, as in the case they relate quantities in units of the same dimension, even if their units of measurement are initially different. For example, the ratio can be reduced by changing the first value to 60 seconds, so the ratio becomes . Once the units are the same, they can be omitted, and the ratio can be reduced to 3∶2. On the other hand, there are non- dimensionless ratios, also known as rates.
|
|
In fluid dynamics, the Darcy friction factor formulae are equations that allow the calculation of the Darcy friction factor, a dimensionless quantity used in the Darcy–Weisbach equation, for the description of friction losses in pipe flow as well as open-channel flow. The Darcy friction factor is also known as the Darcy–Weisbach friction factor, resistance coefficient or simply friction factor; by definition it is four times larger than the Fanning friction factor., 420 pages. See page 293.
|
|
Magnification and demagnification by lenses and other elements can cause a relatively large stop to be the aperture stop for the system. In astrophotography, the aperture may be given as a linear measure (for example in inches or mm) or as the dimensionless ratio between that measure and the focal length. In other photography, it is usually given as a ratio. Sometimes stops and diaphragms are called apertures, even when they are not the aperture stop of the system.
|
|
In doing so, they employed a dimensionless energy-based parameter that relates the contact depth to the maximum depth of penetration. For a soft material, the difference between the contact depth and the maximum depth of penetration is small, and hence its nominal and true hardness values are practically the same. For a harder material these two types of hardness are remarkably different as the difference between them is relatively large. The model proposed by Jha et al.
|
|
In the mathematical theory of probability, offered load is a concept in queuing theory. The offered load is a measure of traffic in a queue. The offered load is given by Little's law: the arrival rate into the queue (symbolized with λ) multiplied by the mean holding time (symbolized by τ), equals the average amount of time spent by items in the queue. Offered load is expressed in Erlang units or call-seconds per hour, a dimensionless measure.
|
|
The United States Air Force Stability and Control Digital DATCOM is a computer program that implements the methods contained in the USAF Stability and Control DATCOM to calculate the static stability, control and dynamic derivative characteristics of fixed-wing aircraft. Digital DATCOM requires an input file containing a geometric description of an aircraft, and outputs its corresponding dimensionless stability derivatives according to the specified flight conditions. The values obtained can be used to calculate meaningful aspects of flight dynamics.
|
|
The particle number (or number of particles) of a thermodynamic system, conventionally indicated with the letter N, is the number of constituent particles in that system. The particle number is a fundamental parameter in thermodynamics which is conjugate to the chemical potential. Unlike most physical quantities, particle number is a dimensionless quantity. It is an extensive parameter, as it is directly proportional to the size of the system under consideration, and thus meaningful only for closed systems.
|
|
The "luminance contrast" is the ratio between the higher luminance, LH, and the lower luminance, LL, that define the feature to be detected. This ratio, often called contrast ratio, CR, (actually being a luminance ratio), is often used for high luminances and for specification of the contrast of electronic visual display devices. The luminance contrast (ratio), CR, is a dimensionless number, often indicated by adding ":1" to the value of the quotient (e.g. CR = 900:1).
|
|
In the original paper of Coleman-Weinberg, as well as in the thesis of Erick Weinberg, Coleman and Weinberg discussed the renormalization of the couplings in various theories, and introduced the concept of "dimensional transmutation"---the running of coupling constants renders some coupling determined by an arbitrary energy scale, therefore although classically one starts from a theory in which there are several arbitrary dimensionless constants, one ends up with a theory with an arbitrary dimensionful parameter.
|
|
The Grashof number (Gr) is a dimensionless number in fluid dynamics and heat transfer which approximates the ratio of the buoyancy to viscous force acting on a fluid. It frequently arises in the study of situations involving natural convection and is analogous to the Reynolds number. It's believed to be named after Franz Grashof. Though this grouping of terms had already been in use, it wasn't named until around 1921, 28 years after Franz Grashof's death.
|
|
Although API gravity is mathematically a dimensionless quantity (see the formula below), it is referred to as being in 'degrees'. API gravity is graduated in degrees on a hydrometer instrument. API gravity values of most petroleum liquids fall between 10 and 70 degrees. In 1916, the U.S. National Bureau of Standards accepted the Baumé scale, which had been developed in France in 1768, as the U.S. standard for measuring the specific gravity of liquids less dense than water.
|
|
Bacteria, particles, organic and inorganic sources of contamination vary depending on a number of factors including the feed water to make UPW as well as the selection of the piping materials to convey it. Bacteria are typically reported in colony- forming units (CFU) per volume of UPW. Particles use number per volume of UPW. Total organic carbon (TOC), metallic contaminants, and anionic contaminants are measured in dimensionless terms of parts per notation, such as ppm, ppb, ppt and ppq.
|
|
Eddington may have been the first to attempt in vain to derive the basic dimensionless constants from fundamental theories and equations, but he was certainly not the last. Many others would subsequently undertake similar endeavors, and efforts occasionally continue even today. None have yet produced convincing results or gained wide acceptance among theoretical physicists. The mathematician Simon Plouffe has made an extensive search of computer databases of mathematical formulae, seeking formulae for the mass ratios of the fundamental particles.
|
|
This equation has the same structure as the one for the point monopole shown above. Although the amplitude factor A replaces the dimensionless volumetric flow rate in these equations, the speed dependence strongly confirms the monopole-like characteristics of the Hartmann whistle. A cousin to the Hartmann whistle is shown in the figure on the right, the Galton whistle. Here the cavity is excited by an annular jet, which oscillates symmetrically around the sharp edges of the cavity.
|
|
Rschevkin, S. N., "The Theory of Sound", The MacMillan Company, 1963. The frequencies were Strouhal number scaled with U, and the sound levels were scaled with the dipole sound power rule of U6 over a speed range of 3 to 1. The data fit was quite good, confirming dynamic similarity and the dipole model. The slight discrepancy in level and frequency overlap suggests that both the dimensionless force and the Strouhal number had weak dependence on the Reynolds number.
|
|
If some particular physical constant had changed, how would we notice it, or how would physical reality be different? Which changed constants result in a meaningful and measurable difference in physical reality? If a physical constant that is not dimensionless, such as the speed of light, did in fact change, would we be able to notice it or measure it unambiguously? – a question examined by Michael Duff in his paper "Comment on time-variation of fundamental constants".
|
|
The SI unit of spin is the (N·m·s) or (kg·m2·s−1), just as with classical angular momentum. In practice, spin is given as a dimensionless spin quantum number by dividing the spin angular momentum by the reduced Planck constant , which has the same dimensions as angular momentum, although this is not the full computation of this value. Very often, the "spin quantum number" is simply called "spin". The fact that it is a quantum number is implicit.
|
|
Surfing on shoaling and breaking waves. The phase velocity cp (blue) and group velocity cg (red) as a function of water depth h for surface gravity waves of constant frequency, according to Airy wave theory. Quantities have been made dimensionless using the gravitational acceleration g and period T, with the deep-water wavelength given by L0 = gT2/(2π) and the deep-water phase speed c0 = L0/T. The grey line corresponds with the shallow- water limit cp =cg = √(gh).
|
|
Further it is discussed that said projectile's weight was one pound.Ingalls, James M., Exterior Ballistics in the Plan Fire, 1886; p. 15, D. Van Nostrand Publisher For the purposes of mathematical convenience for any standard projectile (G) the BC is 1.00. Where as the projectile's sectional density (SD) is dimensionless with a mass of 1 divided by the square of the diameter of 1 caliber equaling an SD of 1. Then the standard projectile is assigned a coefficient of form of 1.
|
|
M–O similarity theory further generalizes the mixing length theory in non-neutral conditions by using so-called "universal functions" of dimensionless height to characterize vertical distributions of mean flow and temperature. The Obukhov length (L), a characteristic length scale of surface layer turbulence derived by Obukhov in 1946, is used for non-dimensional scaling of the actual height. M–O similarity theory marked a significant landmark of modern micrometeorology, providing a theoretical basis for micrometerological experiments and measurement techniques.
|
|
The turbulence results from differences in the fluid's speed and direction, which may sometimes intersect or even move counter to the overall direction of the flow (eddy currents). These eddy currents begin to churn the flow, using up energy in the process, which for liquids increases the chances of cavitation. Reynolds numbers are an important dimensionless quantity in fluid mechanics. The Reynolds number has wide applications, ranging from liquid flow in a pipe to the passage of air over an aircraft wing.
|
|
The point at which this happened was the transition point from laminar to turbulent flow. From these experiments came the dimensionless Reynolds number for dynamic similarity—the ratio of inertial forces to viscous forces. Reynolds also proposed what is now known as the Reynolds-averaging of turbulent flows, where quantities such as velocity are expressed as the sum of mean and fluctuating components. Such averaging allows for 'bulk' description of turbulent flow, for example using the Reynolds- averaged Navier–Stokes equations.
|
|
The COP is a ratio with the same metric units of energy (joules) in both the numerator and denominator. They cancel out, leaving a dimensionless quantity. Formulas for the approximate conversion between SEER and EER or COP are available. : (1) SEER = EER ÷ 0.9 : (2) SEER = COP × 3.792 : (3) EER = COP × 3.413 From equation (2) above, a SEER of 13 is equivalent to a COP of 3.43, which means that 3.43 units of heat energy are pumped per unit of work energy.
|
|
Partial directive gain is the power density in a particular direction and for a particular component of the polarization, divided by the average power density for all directions and all polarizations. For any pair of orthogonal polarizations (such as left-hand- circular and right-hand-circular), the individual power densities simply add to give the total power density. Thus, if expressed as dimensionless ratios rather than in dB, the total directive gain is equal to the sum of the two partial directive gains.
|
|
Rostami B., Kharrat R., Ghotbi S., Darvish F., Gas-Oil Relative Permeability and Residual Oil Saturation as Related to Displacement Instability and Dimensionless numbers, Oil & Gas Science and Technology-Revue de l'IFP, Vol. 65 (2010), No. 2, pp. 299–313. 55\. Dehghan, A.A., Farzaneh, S.A., Kharrat, R., Ghazanfari, M.: Pore level investigation of heavy oil recovery during water alternating solvent injection processes, accepted for published in journal of Trans. Porous media, Aug, DOI 10.1007/11242-009-9463-5, (2010) 83:653–666. 56\.
|
|
The Dynamic Response Index (DRI) is a measure of the likelihood of spinal damage arising from a vertical shock load such as might be encountered in a military environment (i.e., during a mine blast, or in an ejection seat). The DRI is a dimensionless number which is proportional to the maximum spinal compression suffered during the event. The DRI is derived as the solution to an equation which models the human spine as a lumped single-degree-of-freedom spring-shock absorber system.
|
|
The g-factor is a dimensionless factor associated to the nuclear magnetic moment. This parameter contains the sign of the nuclear magnetic moment, which is very important in nuclear structure since it provides information about which type of nucleon (proton or neutron) is dominating over the nuclear wave function. The positive sign is associated to the proton domination and the negative sign with the neutron domination. The values of g(l) and g(s) are known as the g-factors of the nucleons.
|
|
The Ekman number (Ek) is a dimensionless number used in fluid dynamics to describe the ratio of viscous forces to Coriolis forces. It is frequently used in describing geophysical phenomena in the oceans and atmosphere in order to characterise the ratio of viscous forces to the Coriolis forces arising from planetary rotation. It is named after the Swedish oceanographer Vagn Walfrid Ekman. When the Ekman number is small, disturbances are able to propagate before decaying owing to low frictional effects.
|
|
While a graduate student at the University of Michigan, Langton created the Langton ant and Langton loop, both simple artificial life simulations, in addition to his Lambda parameter, a dimensionless measure of complexity and computation potential in cellular automata, given by a chosen state divided by all the possible states. For a 2-state, 1-r neighborhood, 1D cellular automata the value is close to 0.5. For a 2-state, Moore neighborhood, 2D cellular automata, like Conway's Life, the value is 0.273.
|
|
When the velocity was increased, the layer broke up at a given point and diffused throughout the fluid's cross-section. The point at which this happened was the transition point from laminar to turbulent flow. From these experiments came the dimensionless Reynolds number for dynamic similarity—the ratio of inertial forces to viscous forces. Reynolds also proposed what is now known as Reynolds-averaging of turbulent flows, where quantities such as velocity are expressed as the sum of mean and fluctuating components.
|
|
He shared the prize with Arno Allan Penzias and Robert Woodrow Wilson, who won for discovering the cosmic microwave background. Kapitsa resistance is the thermal resistance (which causes a temperature discontinuity) at the interface between liquid helium and a solid. The Kapitsa–Dirac effect is a quantum mechanical effect consisting of the diffraction of electrons by a standing wave of light. In fluid dynamics, the Kapitza number is a dimensionless number characterizing the flow of thin films of fluid down an incline.
|
|
Electronegativity cannot be directly measured and must be calculated from other atomic or molecular properties. Several methods of calculation have been proposed, and although there may be small differences in the numerical values of the electronegativity, all methods show the same periodic trends between elements. The most commonly used method of calculation is that originally proposed by Linus Pauling. This gives a dimensionless quantity, commonly referred to as the Pauling scale (χr), on a relative scale running from 0.79 to 3.98 (hydrogen = 2.20).
|
|
In creation from chaos myth, initially there is nothing but a formless, shapeless expanse. In these stories the word "chaos" means "disorder", and this formless expanse, which is also sometimes called a void or an abyss, contains the material with which the created world will be made. Chaos may be described as having the consistency of vapor or water, dimensionless, and sometimes salty or muddy. These myths associate chaos with evil and oblivion, in contrast to "order" (cosmos) which is the good.
|
|
Eddington believed he had identified an algebraic basis for fundamental physics, which he termed "E-numbers" (representing a certain group – a Clifford algebra). These in effect incorporated spacetime into a higher-dimensional structure. While his theory has long been neglected by the general physics community, similar algebraic notions underlie many modern attempts at a grand unified theory. Moreover, Eddington's emphasis on the values of the fundamental constants, and specifically upon dimensionless numbers derived from them, is nowadays a central concern of physics.
|
|
In convex geometry, the Mahler volume of a centrally symmetric convex body is a dimensionless quantity that is associated with the body and is invariant under linear transformations. It is named after German-English mathematician Kurt Mahler. It is known that the shapes with the largest possible Mahler volume are the balls and solid ellipsoids; this is now known as the Blaschke–Santaló inequality. The still-unsolved Mahler conjecture states that the minimum possible Mahler volume is attained by a hypercube.
|
|
The dimensionless magnetic Reynolds number, R_m, is also used in cases where there is no physical fluid involved. :R_m = \mu \sigma × (characteristic length) × (characteristic velocity) ::where ::\mu is the magnetic permeability ::\sigma is the electrical conductivity. For R_m < 1 the skin effect is negligible and the eddy current braking torque follows the theoretical curve of an induction motor. For R_m > 30 the skin effect dominates and the braking torque decreases much slower with increasing speed than predicted by the induction motor model.
|
|
In a quantum field theory with a dimensionless coupling g, if g is much less than 1, the theory is said to be weakly coupled. In this case, it is well described by an expansion in powers of g, called perturbation theory. If the coupling constant is of order one or larger, the theory is said to be strongly coupled. An example of the latter is the hadronic theory of strong interactions (which is why it is called strong in the first place).
|
|
Allometric study of locomotion involves the analysis of the relative sizes, masses, and limb structures of similarly shaped animals and how these features affect their movements at different speeds. Patterns are identified based on dimensionless Froude numbers, which incorporate measures of animals’ leg lengths, speed or stride frequency, and weight. Alexander incorporates Froude-number analysis into his “dynamic similarity hypothesis” of gait patterns. Dynamically similar gaits are those between which there are constant coefficients that can relate linear dimensions, time intervals, and forces.
|
|
The amplitude of the resonant peak and the bandwidth of resonance is dependent on the damping conditions and is quantified by the dimensionless quantity Q factor. Higher resonant modes and resonant modes at different planes (transverse, lateral, rotational and flexural) are usually triggered at higher frequencies. The specific frequency vicinity of these modes depends on the nature and boundary conditions of each mechanical system. Additionally, subharmonics, superharmonics or subsuperharmonics of each mode can also be excited at the right boundary conditions.
|
|
Terrestrial Magnetism and Atmospheric Electricity, 43(3), 299-320. It is a dimensionless parameter often used in geophysical exploration to describe the magnetic characteristics of a geological body for help in interpreting magnetic anomaly patterns. The total magnetization of a rock is the sum of its natural remanent magnetization and the magnetization induced by the ambient geomagnetic field. Thus, a Koenigsberger ratio, Q, greater than 1 indicates that the remanence properties contribute the majority of the total magnetization of the rock.
|
|
In fluid dynamics, the Darcy–Weisbach equation is an empirical equation, which relates the head loss, or pressure loss, due to friction along a given length of pipe to the average velocity of the fluid flow for an incompressible fluid. The equation is named after Henry Darcy and Julius Weisbach. The Darcy–Weisbach equation contains a dimensionless friction factor, known as the Darcy friction factor. This is also variously called the Darcy–Weisbach friction factor, friction factor, resistance coefficient, or flow coefficient.
|
|
In 2010, p-type polycrystalline BiCuSeO oxyselenides were reported as possible thermoelectric materials. The weak bonds between the [Cu2Se2]−2 conducting and [Bi2O2]+2 insulating layer, as well as the anharmonic crystal lattice structure, may account for the substance's low thermal conductivity and high thermoelectric performance. Recently, BiCuSeO's ZT value, a dimensionless figure-of-merit indicating thermoelectric performance, has been increased from 0.5 to 1.4. Experiment has shown that Ca doping can improve electrical conductivity, thereby increasing the ZT value.
|
|
Components of the SiGe unicouple Heavily doped semiconductors, such as silicon-germanium (SiGe) thermoelectric couples (also called thermocouples or unicouples), are used in space exploration. SiGe alloys present good thermoelectric properties. Their performance in thermoelectric power production is characterized by high dimensionless figures-of-merit (ZT) under high temperatures, which has been shown to be near 2 in some nanostructured-SiGe models. SiGe alloy devices are mechanically rugged and can withstand severe shock and vibration due to its high tensile strength (i.e.
|
|
One dimensionless parameter characterizing a plasma is the ratio of ion to electron mass. Since this number is large, at least 1836, it is commonly taken to be infinite in theoretical analyses, that is, either the electrons are assumed to be massless or the ions are assumed to be infinitely massive. In numerical studies the opposite problem often appears. The computation time would be intractably large if a realistic mass ratio were used, so an artificially small but still rather large value, for example 100, is substituted.
|
|
Stenotic means narrowed, and a stenotic heart valve is one in which the narrowing of the valve is a result of the plaque formation on the valve. The Womersley number, or alpha parameter, is another dimensionless parameter like the Prandtl number or Reynolds number that has been used in the study of fluid dynamics. This parameter represents a ratio of transient to viscous forces, just as the Reynolds number represented a ratio of inertial to viscous forces. A characteristic frequency represents the time dependence of the parameter.
|
|
He was credited with developing Flammability Index, a dimensionless quantity to assess the flammability of combustible materials. He also worked on plasticization and his studies have assisted in widening the understanding of plasticizers and flame-retardants containing phosphorus. He published several articles in peer-reviewed journals and the online repository of the Indian Academy of Sciences has listed 165 of them. He was associated with the Journal of Applied Polymer Science as a member of their editorial board and sat on a number of councils and committees.
|
|
The render output unit, often abbreviated as "ROP", and sometimes called raster operations pipeline, is a hardware component in modern graphics processing units (GPUs) and one of the final steps in the rendering process of modern graphics cards. The pixel pipelines take pixel (each pixel is a dimensionless point), and texel information and process it, via specific matrix and vector operations, into a final pixel or depth value. This process is called rasterization. So ROPs control antialiasing, when more than one sample is merged into one pixel.
|
|
Because it is a dimensionless ratio, laypeople find it difficult to interpret Sharpe ratios of different investments. For example, how much better is an investment with a Sharpe ratio of 0.5 than one with a Sharpe ratio of -0.2? This weakness was well addressed by the development of the Modigliani risk- adjusted performance measure, which is in units of percent return – universally understandable by virtually all investors. In some settings, the Kelly criterion can be used to convert the Sharpe ratio into a rate of return.
|
|
Well-known examples are the indication of the earthquake strength using the Richter scale, the pH value, as a measure for the acidic or basic character of an aqueous solution or of loudness in decibels . In this case, the negative decimal logarithm of the LD50 values, which is standardized in kg per kg body weight, is considered. : − log10LD50 (kg/kg) = value The dimensionless value found can be entered in a toxin scale. Water as the baseline substance is neatly 1 in the negative logarithmic toxin scale.
|
|
Cline, Donna, Exterior Ballistics Explained, Trajectories, Part 3 "Atmosphere" The Point-Mass Trajectory: The Siacci Method Ballistic Coefficient, 2002; pg 40, Lattie Stone Ballistics The general form for the calculations of trajectory adopted for the G model is the Siacci method. The standard model projectile is a "fictitious projectile" used as the mathematical basis for the calculation of actual projectile's trajectory when an initial velocity is known. The G1 model projectile adopted is in dimensionless measures of 2 caliber radius ogival-head and 3.28 caliber in length.
|
|
In a paper published in 2007 it was claimed that the axial dimensionless moment of inertia coefficient was 0.4. Such a value indicated that Rhea had an almost homogeneous interior (with some compression of ice in the center) while the existence of a rocky core would imply a moment of inertia of about 0.34. In the same year another paper claimed the moment of inertia was about 0.37. Rhea being either partially or fully differentiated would be consistent with the observations of the Cassini probe.
|
|
This decomposition is quite general and all terms are dimensionless. Kaluza then applies the machinery of standard general relativity to this metric. The field equations are obtained from five-dimensional Einstein equations, and the equations of motion from the five-dimensional geodesic hypothesis. The resulting field equations provide both the equations of general relativity and of electrodynamics; the equations of motion provide the four-dimensional geodesic equation and the Lorentz force law, and one finds that electric charge is identified with motion in the fifth dimension.
|
|
This is incorrect because the unit "%" can only be used for dimensionless quantities. Instead, the concentration should simply be given in units of g/mL. Percent solution or percentage solution are thus terms best reserved for mass percent solutions (m/m, m%, or mass solute/mass total solution after mixing), or volume percent solutions (v/v, v%, or volume solute per volume of total solution after mixing). The very ambiguous terms percent solution and percentage solutions with no other qualifiers continue to occasionally be encountered.
|
|
In celestial mechanics, the eccentricity vector of a Kepler orbit is the dimensionless vector with direction pointing from apoapsis to periapsis and with magnitude equal to the orbit's scalar eccentricity. For Kepler orbits the eccentricity vector is a constant of motion. Its main use is in the analysis of almost circular orbits, as perturbing (non-Keplerian) forces on an actual orbit will cause the osculating eccentricity vector to change continuously. For the eccentricity and argument of periapsis parameters, eccentricity zero (circular orbit) corresponds to a singularity.
|
|
In probability theory and statistics, the geometric standard deviation (GSD) describes how spread out are a set of numbers whose preferred average is the geometric mean. For such data, it may be preferred to the more usual standard deviation. Note that unlike the usual arithmetic standard deviation, the geometric standard deviation is a multiplicative factor, and thus is dimensionless, rather than having the same dimension as the input values. Thus, the geometric standard deviation may be more appropriately called geometric SD factor GraphPad GuideKirkwood, T.B.L. (1993).
|
|
The scale ratio of a model represents the proportional ratio of a linear dimension of the model to the same feature of the original. Examples include a 3-dimensional scale model of a building or the scale drawings of the elevations or plans of a building. In such cases the scale is dimensionless and exact throughout the model or drawing. The scale can be expressed in four ways: in words (a lexical scale), as a ratio, as a fraction and as a graphical (bar) scale.
|
|
Demography and quantitative epidemiology are statistical fields that deal with counts or proportions of people, or rates of change in these. Counts and proportions are technically dimensionless, and so have no units of measurement, although identifiers such as "people", "births", "infections" and the like are used for clarity. Rates of change are counts per unit of time and strictly have inverse time dimensions (per unit of time). In demography and epidemiology expressions such as "deaths per year" are used to clarify what is being measured.
|
|
In contrast, electromagnetic waves require no medium, but can still travel through one. One important property of mechanical waves is that their amplitudes are measured in an unusual way, displacement divided by (reduced) wavelength. When this gets comparable to unity, significant nonlinear effects such as harmonic generation may occur, and, if large enough, may result in chaotic effects. For example, waves on the surface of a body of water break when this dimensionless amplitude exceeds 1, resulting in a foam on the surface and turbulent mixing.
|
|
Such interaction between the light and free electrons is called Thomson scattering or linear Thomson scattering. The relative strength of the electromagnetic interaction between two charged particles, such as an electron and a proton, is given by the fine- structure constant. This value is a dimensionless quantity formed by the ratio of two energies: the electrostatic energy of attraction (or repulsion) at a separation of one Compton wavelength, and the rest energy of the charge. It is given by α ≈ , which is approximately equal to .
|
|
Both devices mimicked the form and function of a water strider by incorporating a rowing motion of one pair of legs to propel the device, however one was powered with elastic energy and the other was powered by electrical energy. This research compared the various biomimetic devices to their natural counterparts by showing the difference between many physical and dimensionless parameters. This research could one day lead to small, energy efficient water walking robots that could be used to clean up spills in waterways.
|
|
While it is normally asserted that, as the ratio of two lengths, the radian is a "pure number", although Mohr and Phillips dispute this assertion. However, in mathematical writing, the symbol "rad" is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and when degrees are meant, the degree sign is used. The radian is defined as 1.ISO 80000-3:2006 There is controversy as to whether it is satisfactory in the SI to consider angles to be dimensionless.
|
|
In physics and engineering, the Fourier number (Fo) or Fourier modulus, named after Joseph Fourier, is a dimensionless number that characterizes transient heat conduction. Conceptually, it is the ratio of diffusive or conductive transport rate to the quantity storage rate, where the quantity may be either heat (thermal energy) or matter (particles). The number derives from non- dimensionalization of the heat equation (also known as Fourier's Law) or Fick's second law and is used along with the Biot number to analyze time dependent transport phenomena.
|
|
One such dimensionless physical constant is the fine-structure constant. There are some experimental physicists who assert they have in fact measured a change in the fine structure constant and this has intensified the debate about the measurement of physical constants. According to some theorists there are some very special circumstances in which changes in the fine-structure constant can be measured as a change in dimensionful physical constants. Others however reject the possibility of measuring a change in dimensionful physical constants under any circumstance.
|
|
The Bulk Richardson Number (BRN) is an approximation of the Gradient Richardson number. The BRN is a dimensionless ratio in meteorology related to the consumption of turbulence divided by the shear production (the generation of turbulence kinetic energy caused by wind shear) of turbulence. It is used to show dynamic stability and the formation of turbulence. The BRN is used frequently in meteorology due to widely available rawinsonde (frequently called radiosonde) data and numerical weather forecasts that supply wind and temperature measurements at discrete points in space.
|
|
While these similarity transformations capture some basic properties of plasmas, not all plasma phenomena scale in this way. Consider, for example, the degree of ionization, which is dimensionless and thus would ideally remain unchanged when the system is scaled. The number of charged particles per unit volume is proportional to the current density, which scales as x−2, whereas the number of neutral particles per unit volume scales as x−1 in this transformation, so the degree of ionization does not remain unchanged but scales as x−1.
|
|
In fluid dynamics the Eötvös number (Eo), also called the Bond number (Bo), is a dimensionless number measuring the importance of gravitational forces compared to surface tension forces and is used (together with Morton number) to characterize the shape of bubbles or drops moving in a surrounding fluid. The two names commemorate the Hungarian physicist Loránd Eötvös (1848–1919) and the English physicist Wilfrid Noel Bond (1897–1937), respectively. The term Eötvös number is more frequently used in Europe, while Bond number is commonly used in other parts of the world.
|
|
A uniform electron gas at zero temperature is characterised by a single dimensionless parameter, the so-called Wigner-Seitz radius rs = a / ab, where a is the average inter-particle spacing and ab is the Bohr radius. The kinetic energy of an electron gas scales as 1/rs2, this can be seen for instance by considering a simple Fermi gas. The potential energy, on the other hand, is proportional to 1/rs. When rs becomes larger at low density, the latter becomes dominant and forces the electrons as far apart as possible.
|
|
Rayleigh provided the first theoretical treatment of the elastic scattering of light by particles much smaller than the light's wavelength, a phenomenon now known as "Rayleigh scattering", which notably explains why the sky is blue. He studied and described transverse surface waves in solids, now known as "Rayleigh waves". He contributed extensively to fluid dynamics, with concepts such as the Rayleigh number (a dimensionless number associated with natural convection), Rayleigh flow, the Rayleigh–Taylor instability, and Rayleigh's criterion for the stability of Taylor–Couette flow. He also formulated the circulation theory of aerodynamic lift.
|
|
Hawking is cautiously optimistic that such a unified theory of the Universe may be found soon, in spite of significant challenges. At the time the book was written, "superstring theory" had emerged as the most popular theory of quantum gravity, but this theory and related string theories were still incomplete and had yet to be proven in spite of significant effort (this remains the case as of 2020). String theory proposes that particles behave like one-dimensional "strings", rather than as dimensionless particles as they do in QFT. These strings "vibrate" in many dimensions.
|
|
Shape factors are dimensionless quantities used in image analysis and microscopy that numerically describe the shape of a particle, independent of its size. Shape factors are calculated from measured dimensions, such as diameter, chord lengths, area, perimeter, centroid, moments, etc. The dimensions of the particles are usually measured from two-dimensional cross- sections or projections, as in a microscope field, but shape factors also apply to three-dimensional objects. The particles could be the grains in a metallurgical or ceramic microstructure, or the microorganisms in a culture, for example.
|
|
In fluid dynamics, the Iribarren number or Iribarren parameter – also known as the surf similarity parameter and breaker parameter – is a dimensionless parameter used to model several effects of (breaking) surface gravity waves on beaches and coastal structures. The parameter is named after the Spanish engineer Ramón Iribarren Cavanillas (1900–1967), who introduced it to describe the occurrence of wave breaking on sloping beaches. For instance, the Iribarren number is used to describe breaking wave types on beaches; or wave run-up on – and reflection by – beaches, breakwaters and dikes.
|
|
In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. It is equal to the pressure of an ideal gas which has the same temperature and molar Gibbs free energy as the real gas. Fugacities are determined experimentally or estimated from various models such as a Van der Waals gas that are closer to reality than an ideal gas. The real gas pressure and fugacity are related through the dimensionless fugacity coefficient .
|
|
These must be modeled by more complex equations of state. The deviation from the ideal gas behavior can be described by a dimensionless quantity, the compressibility factor, . The ideal gas model has been explored in both the Newtonian dynamics (as in "kinetic theory") and in quantum mechanics (as a "gas in a box"). The ideal gas model has also been used to model the behavior of electrons in a metal (in the Drude model and the free electron model), and it is one of the most important models in statistical mechanics.
|
|
BMI prime, a modification of the BMI system, is the ratio of actual BMI to upper limit optimal BMI (currently defined at 25 kg/m2), i.e., the actual BMI expressed as a proportion of upper limit optimal. The ratio of actual body weight to body weight for upper limit optimal BMI (25 kg/m2) is equal to BMI Prime. BMI Prime is a dimensionless number independent of units. Individuals with BMI Prime less than 0.74 are underweight; those with between 0.74 and 1.00 have optimal weight; and those at 1.00 or greater are overweight.
|
|
The most common factors include substrates, products and effectors. The scaling of the coefficient ensures that it is dimensionless and independent of the units used to measure the reaction rate and magnitude of the factor. The elasticity coefficient is an integral part of metabolic control analysis and was introduced in the early 1970s and possibly earlier by Henrik Kacser and Burns in Edinburgh and Heinrich and Rapoport in Berlin. The elasticity concept has also been described by other authors, most notably Savageau in Michigan and Clarke at Edmonton.
|
|
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 . 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 1 lbf = 1 lb⋅ft/s2.
|
|
The majority of Feist's works are part of The Riftwar Cycle, and feature the worlds of Midkemia and Kelewan. Human magicians and other creatures on the two planets are able to create rifts through dimensionless space that can connect planets in different solar systems. The novels and short stories of The Riftwar Universe record the adventures of various people on these worlds. Midkemia was originally created as an alternative to the Dungeons & Dragons role-playing game, by Feist and his friends studying at the University of California San Diego.
|
|
Virtual photons in some calculations in quantum field theory may also travel at a different speed for short distances; however, this doesn't imply that anything can travel faster than light. While it has been claimed (see VSL criticism below) that no meaning can be ascribed to a dimensional quantity such as the speed of light varying in time (as opposed to a dimensionless number such as the fine structure constant), in some controversial theories in cosmology, the speed of light also varies by changing the postulates of special relativity.
|
|
The model then uses birth, death, immigration, extinction and speciation to modify community composition over time. ;Hubbell's theta The UNTB model produces a dimensionless "fundamental biodiversity" number, θ, which is derived using the formula: : θ = 2Jmv where: : Jm is the size of the metacommunity (the outside source of immigrants to the local community) :v is the speciation rate in the model Relative species abundances in the UNTB model follow a zero-sum multinomial distribution.Hubbell, S. P.; Lake J. 2003. "The neutral theory of biogeography and biodiversity: and beyond".
|
|
The term fundamental physical constant is sometimes used to refer to universal-but-dimensioned physical constants such as those mentioned above. NIST Increasingly, however, physicists only use fundamental physical constant for dimensionless physical constants, such as the fine-structure constant α. Physical constant, as discussed here, should not be confused with other quantities called "constants", which are assumed to be constant in a given context without being fundamental, such as the "time constant" characteristic of a given system, or material constants (e.g., Madelung constant, electrical resistivity, and heat capacity).
|
|
Root-mean-square sound pressure being obtained with a standard frequency weighting and standard time weighting. The reference pressure is set by International agreement to be 20 micropascals for airborne sound. It follows that the decibel is, in a sense, not a unit, it is simply a dimensionless ratio; in this case the ratio of two pressures. An exponentially averaging sound level meter, which gives a snapshot of the current noise level, is of limited use for hearing damage risk measurements; an integrating or integrating-averaging meter is usually mandated.
|
|
RI measurements are usually reported at a reference temperature of 20 degrees Celsius, which is equal to 68 degrees Fahrenheit, and considered to be room temperature. A reference wavelength of 589.3 nm (the sodium D line) is most often used. Though RI is a dimensionless quantity, it is typically reported as nD20 (or n), where the "n" represents refractive index, the "D" denotes the wavelength, and the 20 denotes the reference temperature. Therefore, the refractive index of water at 20 degrees Celsius, taken at the Sodium D Line, would be reported as 1.3330 nD20.
|
|
In thermodynamics, the reduced properties of a fluid are a set of state variables scaled by the fluid's state properties at its critical point. These dimensionless thermodynamic coordinates, taken together with a substance's compressibility factor, provide the basis for the simplest form of the theorem of corresponding states. Reduced properties are also used to define the Peng–Robinson equation of state, a model designed to provide reasonable accuracy near the critical point. They are also used to critical exponents, which describe the behaviour of physical quantities near continuous phase transitions.
|
|
The essence of asymptotic safety is the observation that nontrivial renormalization group fixed points can be used to generalize the procedure of perturbative renormalization. In an asymptotically safe theory the couplings do not need to be small or tend to zero in the high energy limit but rather tend to finite values: they approach a nontrivial UV fixed point. The running of the coupling constants, i.e. their scale dependence described by the renormalization group (RG), is thus special in its UV limit in the sense that all their dimensionless combinations remain finite.
|
|
" Stephen Hill, co- founder, Hearts of Space, essay titled New Age Music Made Simple and sensations of floating, cruising or flying."...Spacemusic ... conjures up either outer "space" or "inner space" " – Lloyd Barde, founder of Backroads Music Notes on Ambient Music, Hyperreal Music Archive "Space And Travel Music: Celestial, Cosmic, and Terrestrial... This New Age sub-category has the effect of outward psychological expansion. Celestial or cosmic music removes listeners from their ordinary acoustical surroundings by creating stereo sound images of vast, virtually dimensionless spatial environments. In a word — spacey.
|
|
There is another measure of the relationship between two random variables that is often easier to interpret than the covariance. The correlation just scales the covariance by the product of the standard deviation of each variable. Consequently, the correlation is a dimensionless quantity that can be used to compare the linear relationships between pairs of variables in different units. If the points in the joint probability distribution of X and Y that receive positive probability tend to fall along a line of positive (or negative) slope, ρXY is near +1 (or −1).
|
|
The thrust-to- weight ratio can be calculated by dividing the thrust (in SI units - in newtons) by the weight (in newtons) of the engine or vehicle and is a dimensionless quantity. Note that the thrust can also be measured in pound- force (lbf) provided the weight is measured in pounds (lb); the division of these two values still gives the numerically correct thrust-to-weight ratio. For valid comparison of the initial thrust-to-weight ratio of two or more engines or vehicles, thrust must be measured under controlled conditions.
|
|
In dimensional analysis, the Strouhal number (St, or sometimes Sr to avoid the conflict with the Stanton number) is a dimensionless number describing oscillating flow mechanisms. The parameter is named after Vincenc Strouhal, a Czech physicist who experimented in 1878 with wires experiencing vortex shedding and singing in the wind.Strouhal, V. (1878) "Ueber eine besondere Art der Tonerregung" (On an unusual sort of sound excitation), Annalen der Physik und Chemie, 3rd series, 5 (10) : 216–251. The Strouhal number is an integral part of the fundamentals of fluid mechanics.
|
|
In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle from that point. In the International System of Units (SI), a solid angle is expressed in a dimensionless unit called a steradian (symbol: sr).
|
|
Einstein, letter to Felix Klein, 1917. (On determinism and approximations.) Quoted in Pais (1982), Ch. 17. Following this view, we may reasonably hope for a theory of everything which self-consistently incorporates all currently known forces, but we should not expect it to be the final answer. On the other hand, it is often claimed that, despite the apparently ever-increasing complexity of the mathematics of each new theory, in a deep sense associated with their underlying gauge symmetry and the number of dimensionless physical constants, the theories are becoming simpler.
|
|
Since strain is a dimensionless quantity, the units of λ will be the same as the units of stress. Specifying how stress and strain are to be measured, including directions, allows for many types of elastic moduli to be defined. The three primary ones are: # Young's modulus (E) describes tensile elasticity, or the tendency of an object to deform along an axis when opposing forces are applied along that axis; it is defined as the ratio of tensile stress to tensile strain. It is often referred to simply as the elastic modulus.
|
|
Photographic quantity (also known as photoquantity) is a measure of the amount of light received by a sensor, such as a camera, in dimensionless units that account for information lost by integration over the spectral response of the sensor, while otherwise preserving the linear relationship involved in the interaction of light through one or more exposures. The photoquantity is neither radiometric nor photometric. The photoquantity is not radiometric because the sensor, camera, or the like, is not an ideal receiving antenna. Rather, the sensor has some non-flat spectral response.
|
|
There is no exhaustive list of such constants but it does make sense to ask about the minimal number of fundamental constants necessary to determine a given physical theory. Thus, the Standard Model requires 25 physical constants, about half of them the masses of fundamental particles (which become "dimensionless" when expressed relative to the Planck mass or, alternatively, as coupling strength with the Higgs field along with the gravitational constant).Kuntz, I., Gravitational Theories Beyond General Relativity, (Berlin/Heidelberg: Springer, 2019), pp. 58–61. Fundamental physical constants cannot be derived and have to be measured.
|
|
The Render Output Pipeline is an inherited term, and more often referred to as the render output unit. Its job is to control the sampling of pixels (each pixel is a dimensionless point), so it controls antialiasing, when more than one sample is merged into one pixel. All data rendered has to travel through the ROP in order to be written to the framebuffer, from there it can be transmitted to the display. Therefore, the ROP is where the GPU's output is assembled into a bitmapped image ready for display.
|
|
For redox reactions, the equivalent weight of each reactant supplies or reacts with one mole of electrons (e−) in a redox reaction. Equivalent weight has the dimensions and units of mass, unlike atomic weight, which is dimensionless. Equivalent weights were originally determined by experiment, but (insofar as they are still used) are now derived from molar masses. Additionally, the equivalent weight of a compound can be calculated by dividing the molecular mass by the number of positive or negative electrical charges that result from the dissolution of the compound.
|
|
The Hindmarsh–Rose model of neuronal activity is aimed to study the spiking- bursting behavior of the membrane potential observed in experiments made with a single neuron. The relevant variable is the membrane potential, x(t), which is written in dimensionless units. There are two more variables, y(t) and z(t), which take into account the transport of ions across the membrane through the ion channels. The transport of sodium and potassium ions is made through fast ion channels and its rate is measured by y(t), which is called the spiking variable.
|
|
Chemical plant cost indexes are dimensionless numbers employed to updating capital cost required to erect a chemical plant from a past date to a later time, following changes in the value of money due to inflation and deflation. Since, at any given time, the number of chemical plants is insufficient to use in a preliminary or predesign estimate, cost indexes are handy for a series of management purposes, like long-range planning, budgeting and escalating or de- escalating contract costs.Pintelon, L. & Puyvelde, F. V., 1997. Estimating Plant Construction Costs.
|
|
In physics, a characteristic length is an important dimension that defines the scale of a physical system. Often, such a length is used as an input to a formula in order to predict some characteristics of the system, and it is usually required by the construction of a dimensionless quantity, in the general framework of dimensional analysis and in particular applications such as fluid mechanics. In computational mechanics, a characteristic length is defined to force localization of a stress softening constitutive equation. The length is associated with an integration point.
|
|
Thermal emittance or thermal emissivity is the ratio of the radiant emittance of heat of a specific object or surface to that of a standard black body. Emissivity and emittivity are both dimensionless quantities given in the range of 0 to 1, but emissivity refers to a material property (of a homogeneous material), while emissivity refers to specific samples or objects. For building products, thermal emittance measurements are taken for wavelengths in the infrared. Determining the thermal emittance and solar reflectance of building materials, especially roofing materials, can be very useful for reducing heating and cooling energy costs in buildings.
|
|
Although the International Bureau of Weights and Measures (an international standards organization known also by its French-language initials BIPM) recognizes the use of parts-per notation, it is not formally part of the International System of Units (SI). Note that although "percent" (%) is not formally part of the SI, both the BIPM and the International Organization for Standardization (ISO) take the position that "in mathematical expressions, the internationally recognized symbol % (percent) may be used with the SI to represent the number 0.01" for dimensionless quantities.Quantities and units. Part 0: General principles, ISO 31-0:1992.
|
|
Another problem of the parts-per notation is that it may refer to mass fraction, mole fraction or volume fraction. Since it is usually not stated which quantity is used, it is better to write the unit as kg/kg, mol/mol or m3/m3 (even though they are all dimensionless). The difference is quite significant when dealing with gases, and it is very important to specify which quantity is being used. For example, the conversion factor between a mass fraction of 1 ppb and a mole fraction of 1 ppb is about 4.7 for the greenhouse gas CFC-11 in air.
|
|
There are many dimensionless numbers in fluid mechanics. The Reynolds number measures the ratio of advection and diffusion effects on structures in the velocity field, and is therefore closely related to Péclet numbers, which measure the ratio of these effects on other fields carried by the flow, for example temperature and magnetic fields. Replacement of the kinematic viscosity in by the thermal or magnetic diffusivity results in respectively the thermal Péclet number and the magnetic Reynolds number. These are therefore related to by products with ratios of diffusivities, namely the Prandtl number and magnetic Prandtl number.
|
|
Chekani M. and Kharrat R., An integrated reservoir characterization analysis in a carbonate reservoir: A case study, Petroleum Science and Technology, Vol. 30 (2012), pp 1468–1485. 101\. Bolouri H., Schaffie M, ., Kharrat R., Ghazanfari M.H., Ghoodjani E, An Experimental and Modeling Study of Asphaltene Deposition due to CO2 Miscible Injection, Journal of Petroleum Science and Technology, Volume: 31, Issue: 02, (2012) pages 129 – 141\. 102\. Sadati S. E., Kharrat R., Scaling of Gas Assisted Gravity Drainage Process using Dimensionless Groups, Journal of Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, Volume: 35, Issue: 02, (2012) pages 164 - 172\. 103\.
|
|
The signal intensity was measured as signal-to-noise ratio, with the noise (or baseline) averaged over the previous few minutes. The signal was sampled for 10 seconds and then processed by the computer, which took 2 seconds. Therefore, every 12 seconds the result for each frequency channel was output on the printout as a single character, representing the 10-second average intensity, minus the baseline, expressed as a dimensionless multiple of the signal's standard deviation. In the chosen alphanumeric measuring system, a space character denotes an intensity between 0 and 1, that is between baseline and one standard deviation above it.
|
|
SI units predominate in most fields, and continue to increase in popularity at the expense of Gaussian units. Alternative unit systems also exist. Conversions between quantities in the Gaussian unit system and the SI unit system are not as straightforward as direct unit conversions because the quantities themselves are defined differently in the different systems, which has the effect that the equations expressing physical laws of electromagnetism (such as Maxwell's equations) change depending on what system of units is being used. As an example, quantities that are dimensionless in one system may have dimension in another.
|
|
In biology, the "%" symbol is sometimes incorrectly used to denote mass concentration, also called "mass/volume percentage." A solution with 1 g of solute dissolved in a final volume of 100 mL of solution would be labeled as "1%" or "1% m/v" (mass/volume). The notation is mathematically flawed because the unit "%" can only be used for dimensionless quantities. "Percent solution" or "percentage solution" are thus terms best reserved for "mass percent solutions" (m/m = m% = mass solute/mass total solution after mixing), or "volume percent solutions" (v/v = v% = volume solute per volume of total solution after mixing).
|
|
The Weibull modulus is a dimensionless parameter of the Weibull distribution which is used to describe variability in measured material strength of brittle materials. For ceramics and other brittle materials, the maximum stress that a sample can be measured to withstand before failure may vary from specimen to specimen, even under identical testing conditions. This is related to the distribution of physical flaws present in the surface or body of the brittle specimen, since brittle failure processes originate at these weak points. When flaws are consistent and evenly distributed, samples will behave more uniformly than when flaws are clustered inconsistently.
|
|
Although the value is relative to the standards against which it is compared, the unit used to measure the times changes the score (see examples 1 and 2). This is a consequence of the requirement that the argument of the logarithmic function must be dimensionless. The multiplier also can't have a numeric value of 1 or less, because the logarithm of 1 is 0 (examples 3 and 4), and the logarithm of any value less than 1 is negative (examples 5 and 6); that would result in scores of value 0 (even with changes), undefined, or negative (even if better than positive).
|
|
Brachydactyly can also be a signal that one will be at risk for heart problems as they age. Nomograms for normal values of finger length as a ratio to other body measurements have been published. In clinical genetics the most commonly used index of digit length is the dimensionless ratio of the length of the 3rd (middle) finger to the hand length. Both are expressed in the same units (centimeters, for example) and are measured in an open hand from the fingertip to the principal creases where the finger joins the palm and where the palm joins the wrist.
|
|
The propagation constant of the sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.
|
|
The radian is abbreviated rad, though this symbol is often omitted in mathematical texts, where radians are assumed unless specified otherwise. When radians are used angles are considered as dimensionless. The radian is used in virtually all mathematical work beyond simple practical geometry, due, for example, to the pleasing and "natural" properties that the trigonometric functions display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the SI system. ;Clock position (n = 12): A clock position is the relative direction of an object described using the analogy of a 12-hour clock.
|
|
Although named for Edgar Buckingham, the theorem was first proved by French mathematician Joseph Bertrand in 1878. Bertrand considered only special cases of problems from electrodynamics and heat conduction, but his article contains, in distinct terms, all the basic ideas of the modern proof of the theorem and clearly indicates the theorem's utility for modelling physical phenomena. The technique of using the theorem (“the method of dimensions”) became widely known due to the works of Rayleigh. The first application of the theorem in the general caseWhen in applying the pi–theorem there arises an arbitrary function of dimensionless numbers.
|
|
Incl .. 24.66109 where the epoch is expressed in terms of Terrestrial Time, with an equivalent Julian date. Four of the elements are independent of any particular coordinate system: M is mean anomaly (deg), n: mean daily motion (deg/d), a: size of semi-major axis (AU), e: eccentricity (dimensionless). But the argument of perihelion, longitude of the ascending node and the inclination are all coordinate-dependent, and are specified relative to the reference frame of the equinox and ecliptic of another date "2000.0", otherwise known as J2000, i.e. January 1.5, 2000 (12h on January 1) or JD 2451545.0.
|
|
Thrust-to-weight ratio is a dimensionless ratio of thrust to weight of a rocket, jet engine, propeller engine, or a vehicle propelled by such an engine that is an indicator of the performance of the engine or vehicle. The instantaneous thrust-to-weight ratio of a vehicle varies continually during operation due to progressive consumption of fuel or propellant and in some cases a gravity gradient. The thrust-to-weight ratio based on initial thrust and weight is often published and used as a figure of merit for quantitative comparison of a vehicle's initial performance.
|
|
Carter chose to focus on a tautological aspect of his ideas, which has resulted in much confusion. In fact, anthropic reasoning interests scientists because of something that is only implicit in the above formal definitions, namely that we should give serious consideration to there being other universes with different values of the "fundamental parameters"—that is, the dimensionless physical constants and initial conditions for the Big Bang. Carter and others have argued that life as we know it would not be possible in most such universes. In other words, the universe we are in is fine tuned to permit life.
|
|
As the description implies, Eb is the signal energy associated with each user data bit; it is equal to the signal power divided by the user bit rate (not the channel symbol rate). If signal power is in watts and bit rate is in bits per second, Eb is in units of joules (watt-seconds). N0 is the noise spectral density, the noise power in a 1 Hz bandwidth, measured in watts per hertz or joules. These are the same units as Eb so the ratio Eb/N0 is dimensionless; it is frequently expressed in decibels.
|
|
Relative atomic mass is determined by the average atomic mass, or the weighted mean of the atomic masses of all the atoms of a particular chemical element found in a particular sample, which is then compared to the atomic mass of carbon-12. This comparison is the quotient of the two weights, which makes the value dimensionless (no unit appended). This quotient also explains the word relative: the sample mass value is considered relative to that of carbon-12. It is a synonym for atomic weight, though it is not to be confused with relative isotopic mass.
|
|
These estimates assume that all matter can be taken to be hydrogen and require assumed values for the number and size of galaxies and stars in the universe. Attempts to find a mathematical basis for this dimensionless constant have continued up to the present time. During a course of lectures that he delivered in 1938 as Tarner Lecturer at Trinity College, Cambridge, Eddington averred that: This large number was soon named the "Eddington number". Shortly thereafter, improved measurements of α yielded values closer to 1/137, whereupon Eddington changed his "proof" to show that α had to be exactly 1/137.
|
|
In describing the units of a rate, the word "per" is used to separate the units of the two measurements used to calculate the rate (for example a heart rate is expressed "beats per minute"). A rate defined using two numbers of the same units (such as tax rates) or counts (such as literacy rate) will result in a dimensionless quantity, which can be expressed as a percentage (for example, the global literacy rate in 1998 was 80%), fraction, or multiple. Often rate is a synonym of rhythm or frequency, a count per second (i.e., hertz); e.g.
|
|
In abstract vector spaces, the length of the arrow depends on a dimensionless scale. If it represents, for example, a force, the "scale" is of physical dimension length/force. Thus there is typically consistency in scale among quantities of the same dimension, but otherwise scale ratios may vary; for example, if "1 newton" and "5 m" are both represented with an arrow of 2 cm, the scales are 1 m:50 N and 1:250 respectively. Equal length of vectors of different dimension has no particular significance unless there is some proportionality constant inherent in the system that the diagram represents.
|
|
If the edge tone is relevant, perhaps the characteristic dimension should be the gap between the blades. The researchers deduced that the fluctuating force was proportional to U2, but the sound power was found to vary from U4.5 to U6.0. If the measurement bandwidth is broad and the measurement distance is out of the near field, there are two dynamic factors (Strouhal number and dimensionless force), that can cause the exponent to be less than 6. Both the Deltameter and hole tone data show the Strouhal number is a weak negative function of Reynolds number, which is squared in the sound power equation.
|
|
Atmospheric and oceanographic flows take place over horizontal length scales which are very large compared to their vertical length scale, and so they can be described using the shallow water equations. The Rossby number is a dimensionless number which characterises the strength of inertia compared to the strength of the Coriolis force. The quasi-geostrophic equations are approximations to the shallow water equations in the limit of small Rossby number, so that inertial forces are an order of magnitude smaller than the Coriolis and pressure forces. If the Rossby number is equal to zero then we recover geostrophic flow.
|
|
Instead, these chemists had settled on a list of what were universally called "equivalents" (H = 1, O = 8, C = 6, S = 16, Cl = 35.5, Na = 23, Ca = 20, and so on). However, these nineteenth-century "equivalents" were not equivalents in the original or modern sense of the term. Since they represented dimensionless numbers that for any given element were unique and unchanging, they were in fact simply an alternative set of atomic weights, in which the elements of even valence have atomic weights one- half of the modern values. This fact was not recognized until much later.
|
|
A careful distinction needs to be made between abstract quantities and measurable quantities. The multiplication and division rules of quantity calculus are applied to SI base units (which are measurable quantities) to define SI derived units, including dimensionless derived units, such as the radian (rad) and steradian (sr) which are useful for clarity, although they are both algebraically equal to 1. Thus there is some disagreement about whether it is meaningful to multiply or divide units. Emerson suggests that if the units of a quantity are algebraically simplified, they then are no longer units of that quantity.
|
|
In fluid mechanics, added mass or virtual mass is the inertia added to a system because an accelerating or decelerating body must move (or deflect) some volume of surrounding fluid as it moves through it. Added mass is a common issue because the object and surrounding fluid cannot occupy the same physical space simultaneously. For simplicity this can be modeled as some volume of fluid moving with the object, though in reality "all" the fluid will be accelerated, to various degrees. The dimensionless added mass coefficient is the added mass divided by the displaced fluid mass – i.e.
|
|
Strain means Deformation, and is defined as relative change in length. The Lagrangian formula εL = (L-L0)/L0 = ΔL/L0, where L0 is baseline length and L is the resulting length, defines strain in relation to the original length as a dimensionless measure, where shortening will be negative, and lengthening will be positive. It is usually expressed in percent. An alternative definition, Eulerian strain defines the strain in relation to the instantaneous length: εE = ΔL/L. For a change over time, the Lagrangian strain will be: εL = Σ ΔL/L0, and Eulerian Strain εE = Σ (ΔL/L).
|
|
Trend-following and contrarian patterns are found to coexist and depend on the dimensionless time horizon. Using a renormalisation group approach, the probabilistic based scenario approach exhibits statistically signifificant predictive power in essentially all tested market phases. A survey of modern studies by Park and IrwinC-H Park and S.H. Irwin, "The Profitability of Technical Analysis: A Review" AgMAS Project Research Report No. 2004-04 showed that most found a positive result from technical analysis. In 2011, Caginalp and DeSantisG. Caginalp and M. DeSantis, "Nonlinearity in the dynamics of financial markets," Nonlinear Analysis: Real World Applications, 12(2), 1140-1151, 2011.
|
|
In electrical engineering, the power factor of an AC electrical power system is defined as the ratio of the real power absorbed by the load to the apparent power flowing in the circuit, and is a dimensionless number in the closed interval of −1 to 1. A power factor of less than one indicates the voltage and current are not in phase, reducing the average product of the two. Real power is the instantaneous product of voltage and current and represents the capacity of the electricity for performing work. Apparent power is the product of RMS current and voltage.
|
|
The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.
|
|
In the cases above, gain will be a dimensionless quantity, as it is the ratio of like units (decibels are not used as units, but rather as a method of indicating a logarithmic relationship). In the bipolar transistor example, it is the ratio of the output current to the input current, both measured in amperes. In the case of other devices, the gain will have a value in SI units. Such is the case with the operational transconductance amplifier, which has an open-loop gain (transconductance) in siemens (mhos), because the gain is a ratio of the output current to the input voltage.
|
|
In continuum mechanics, lateral strain, also known as transverse strain, is defined as the ratio of the change in diameter of a circular bar of a material to its diameter due to deformation in the longitudinal direction. It occurs when under the action of a longitudinal stress, a body will extend in the direction of the stress and contract in the transverse or lateral direction (in the case of tensile stress). When put under compression, the body will contract in the direction of the stress and extend in the transverse or lateral direction. It is a dimensionless quantity, as it is a ratio between two quantities of the same dimension.
|
|
When a fluid is flowing through a closed channel such as a pipe or between two flat plates, either of two types of flow may occur depending on the velocity and viscosity of the fluid: laminar flow or turbulent flow. Laminar flow occurs at lower velocities, below a threshold at which the flow becomes turbulent. The velocity is determined by a dimensionless parameter characterizing the flow called the Reynolds number, which also depends on the viscosity and density of the fluid and dimensions of the channel. Turbulent flow is a less orderly flow regime that is characterized by eddies or small packets of fluid particles, which result in lateral mixing.
|
|
His proof intended to show that the ratio was independent of the nature of the non-ideal body, however partly transparent or partly reflective it was. His proof first argued that for wavelength and at temperature , at thermal equilibrium, all perfectly black bodies of the same size and shape have the one and the same common value of emissive power , with the dimensions of power. His proof noted that the dimensionless wavelength-specific absorptivity of a perfectly black body is by definition exactly 1. Then for a perfectly black body, the wavelength-specific ratio of emissive power to absorptivity is again just , with the dimensions of power.
|
|
Drag coefficients in fluids with Reynolds number approximately 104 In fluid dynamics, the drag coefficient (commonly denoted as: c_d, c_x or c_w) is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag equation in which a lower drag coefficient indicates the object will have less aerodynamic or hydrodynamic drag. The drag coefficient is always associated with a particular surface area. The drag coefficient of any object comprises the effects of the two basic contributors to fluid dynamic drag: skin friction and form drag.
|
|
In fluid mechanics, the Rayleigh number (Ra) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free or natural convection. It characterises the fluid's flow regime: a value in a certain lower range denotes laminar flow; a value in a higher range, turbulent flow. Below a certain critical value, there is no fluid motion and heat transfer is by conduction rather than convection. The Rayleigh number is defined as the product of the Grashof number, which describes the relationship between buoyancy and viscosity within a fluid, and the Prandtl number, which describes the relationship between momentum diffusivity and thermal diffusivity.
|
|
Before reading this section, it must be understood that the 2019 redefinition of the SI base units concluded that the molar mass constant is not exactly , but The molar mass of atoms of an element is given by the relative atomic mass of the element multiplied by the molar mass constant, M≈ = 1.000000g/mol. For normal samples from earth with typical isotope composition, the atomic weight can be approximated by the standard atomic weight or the conventional atomic weight. :M(H) = × = :M(S) = × = :M(Cl) = × = :M(Fe) = × = . Multiplying by the molar mass constant ensures that the calculation is dimensionally correct: standard relative atomic masses are dimensionless quantities (i.e.
|
|
Generally, it is the case when the motion of a particle is described in the position space, where the corresponding probability amplitude function is the wave function. If the function represents the quantum state vector , then the real expression , that depends on , forms a probability density function of the given state. The difference of a density function from simply a numerical probability means that one should integrate this modulus-squared function over some (small) domains in to obtain probability values – as was stated above, the system can't be in some state with a positive probability. It gives to both amplitude and density function a physical dimension, unlike a dimensionless probability.
|
|
Isaac Newton (1642–1727) by formulating the laws of motion and his law of viscosity, in addition to developing the calculus, paved the way for many great developments in fluid mechanics. Using Newton's laws of motion, numerous 18th-century mathematicians solved many frictionless (zero-viscosity) flow problems. However, most flows are dominated by viscous effects, so engineers of the 17th and 18th centuries found the inviscid flow solutions unsuitable, and by experimentation they developed empirical equations, thus establishing the science of hydraulics. Late in the 19th century, the importance of dimensionless numbers and their relationship to turbulence was recognized, and dimensional analysis was born.
|
|
The Biot number (Bi) is a dimensionless quantity used in heat transfer calculations. It is named after the eighteenth century French physicist Jean- Baptiste Biot (1774–1862), and gives a simple index of the ratio of the heat transfer resistances inside of a body and at the surface of a body. This ratio determines whether or not the temperatures inside a body will vary significantly in space, while the body heats or cools over time, from a thermal gradient applied to its surface. __TOC__ In general, problems involving small Biot numbers (much smaller than 1) are thermally simple, due to uniform temperature fields inside the body.
|
|
A bell-shaped production curve, as originally suggested by M. King Hubbert in 1956 Hubbert made several contributions to geophysics, including a mathematical demonstration that rock in the earth's crust, because it is under immense pressure in large areas, should exhibit plasticity, similar to clay. This demonstration explained the observed results that the earth's crust deforms over time. He also studied the flow of underground fluids. Based on theoretical arguments, Hubbert (1940) proposed a constitutive equation K_{abs} = N D^{2} for absolute permeability K_{abs} of an underground water or oil reservoir where D is the average grain diameter and N is a dimensionless proportionality constant.
|
|
The gross structure of line spectra is the line spectra predicted by the quantum mechanics of non-relativistic electrons with no spin. For a hydrogenic atom, the gross structure energy levels only depend on the principal quantum number n. However, a more accurate model takes into account relativistic and spin effects, which break the degeneracy of the energy levels and split the spectral lines. The scale of the fine structure splitting relative to the gross structure energies is on the order of (Zα)2, where Z is the atomic number and α is the fine-structure constant, a dimensionless number equal to approximately 1/137.
|
|
An elliptic, parabolic, and hyperbolic Kepler orbit: Elliptic orbit by eccentricity The orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values between 0 and 1 form an elliptic orbit, 1 is a parabolic escape orbit, and greater than 1 is a hyperbola. The term derives its name from the parameters of conic sections, as every Kepler orbit is a conic section. It is normally used for the isolated two-body problem, but extensions exist for objects following a Klemperer rosette orbit through the galaxy.
|
|
In the new SI system, the permittivity of vacuum will not be a constant anymore, but a measured quantity, related to the (measured) dimensionless fine structure constant. Another historical synonym was "dielectric constant of vacuum", as "dielectric constant" was sometimes used in the past for the absolute permittivity. However, in modern usage "dielectric constant" typically refers exclusively to a relative permittivity ε/ε0 and even this usage is considered "obsolete" by some standards bodies in favor of relative static permittivity. Hence, the term "dielectric constant of vacuum" for the electric constant ε0 is considered obsolete by most modern authors, although occasional examples of continuing usage can be found.
|
|
In the Cartesian view, the distinction between these two concepts is a methodological necessity driven by a distrust of the senses and the res extensa as it represents the entire material world. The categorical separation of these two, however, caused a problem, which can be demonstrated in this question: How can a wish (a mental event), cause an arm movement (a physical event)? Descartes has not provided any answer to this but Gottfried Leibniz proposed that it can be addressed by endowing each geometrical point in the res extensa with mind. Each of these points is within res extensa but they are also dimensionless, making them unextended.
|
|
But this would be a mistake. If c, h, and e were all changed so that the values they have in metric (or any other) units were different when we looked them up in our tables of physical constants, but the value of α remained the same, this new world would be observationally indistinguishable from our world. The only thing that counts in the definition of worlds are the values of the dimensionless constants of Nature. If all masses were doubled in value [including the Planck mass mP] you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged.
|
|
As a dimensionless constant which does not seem to be directly related to any mathematical constant, the fine-structure constant has long fascinated physicists. Arthur Eddington argued that the value could be "obtained by pure deduction" and he related it to the Eddington number, his estimate of the number of protons in the universe. This led him in 1929 to conjecture that the reciprocal of the fine-structure constant was not approximately the integer 137, but precisely the integer 137. Other physicists neither adopted this conjecture nor accepted his arguments but by the 1940s experimental values for deviated sufficiently from 137 to refute Eddington's argument.
|
|
Considering walking with the inverted pendulum model, one can predict maximum attainable walking speed with the Froude number, F = v^2 / lg, where v^2 = velocity squared, l = leg length, and g= gravity. The Froude number is a dimensionless value representing the ratio of Centripetal force to Gravitational force during walking. If the body is viewed as a mass moving through a circular arc centered over the foot, the theoretical maximum Froude number is 1.0, where centripetal and gravitational forces are equal. At a number greater than 1.0, the gravitational force would not be strong enough to hold the body in a horizontal plane and the foot would miss the ground.
|
|
The subscript I arises because of the different ways of loading a material to enable a crack to propagate. It refers to so-called "mode I" loading as opposed to mode II or III: The expression for K_I in equation 2.1 will be different for geometries other than the center-cracked infinite plate, as discussed in the article on the stress intensity factor. Consequently, it is necessary to introduce a dimensionless correction factor, Y, in order to characterize the geometry. This correction factor, also often referred to as the geometric shape factor, is given by empirically determined series and accounts for the type and geometry of the crack or notch.
|
|
Dimensional analysis has allowed us to conclude that the period of the pendulum is not a function of its mass. (In the 3D space of powers of mass, time, and distance, we can say that the vector for mass is linearly independent from the vectors for the three other variables. Up to a scaling factor, \vec g + 2 \vec T - \vec L is the only nontrivial way to construct a vector of a dimensionless parameter.) The model can now be expressed as: :f(gT^2/L) = 0. Assuming the zeroes of f are discrete, we can say gT2/L = Cn, where Cn is the nth zero of the function f.
|
|
To nondimensionalize a system of equations, one must do the following: #Identify all the independent and dependent variables; #Replace each of them with a quantity scaled relative to a characteristic unit of measure to be determined; #Divide through by the coefficient of the highest order polynomial or derivative term; #Choose judiciously the definition of the characteristic unit for each variable so that the coefficients of as many terms as possible become 1; #Rewrite the system of equations in terms of their new dimensionless quantities. The last three steps are usually specific to the problem where nondimensionalization is applied. However, almost all systems require the first two steps to be performed.
|
|
The body will in consequence try to move toward the low-pressure zone, in an oscillating movement called vortex-induced vibration. Eventually, if the frequency of vortex shedding matches the natural frequency of the structure, the structure will begin to resonate and the structure's movement can become self-sustaining. The frequency of the vortices in the von Kármán vortex street is called the Strouhal frequency f_s, and is given by Here, stands for the flow velocity, is a characteristic length of the bluff body and is the dimensionless Strouhal number, which depends on the body in question. For Reynolds Numbers greater than 1000, the Strouhal number is approximately equal to 0.21.
|
|
In physics, naturalness is the property that the dimensionless ratios between free parameters or physical constants appearing in a physical theory should take values "of order 1" and that free parameters are not fine-tuned. That is, a natural theory would have parameter ratios with values like 2.34 rather than 234000 or 0.000234. The requirement that satisfactory theories should be "natural" in this sense is a current of thought initiated around the 1960s in particle physics. It is an aesthetic criterion, not a physical one, that arises from the seeming non-naturalness of the standard model and the broader topics of the hierarchy problem, fine-tuning, and the anthropic principle.
|
|
The dimensionless constants that arise in the results obtained, such as the C in the Poiseuille's Law problem and the \kappa in the spring problems discussed above, come from a more detailed analysis of the underlying physics and often arise from integrating some differential equation. Dimensional analysis itself has little to say about these constants, but it is useful to know that they very often have a magnitude of order unity. This observation can allow one to sometimes make "back of the envelope" calculations about the phenomenon of interest, and therefore be able to more efficiently design experiments to measure it, or to judge whether it is important, etc.
|
|
The magnetic Reynolds number has a similar form to both the Péclet number and the Reynolds number. All three can be regarded as giving the ratio of advective to diffusive effects for a particular physical field, and have a similar form of a velocity times a length divided by a diffusivity. The magnetic Reynolds number is related to the magnetic field in an MHD flow, while the Reynolds number is related to the fluid velocity itself, and the Péclet number is related to heat. The dimensionless groups arise in the non-dimensionalization of the respective governing equations, the induction equation, the momentum equation, and the heat equation.
|
|
In nuclear physics and particle physics, isospin (I) is a quantum number related to the strong interaction. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions of baryons and mesons. The name of the concept contains the term spin because its quantum mechanical description is mathematically similar to that of angular momentum (in particular, in the way it couples; for example, a proton-neutron pair can be coupled either in a state of total isospin 1 or in one of 0). But unlike angular momentum it is a dimensionless quantity, and is not actually any type of spin.
|
|
Gustav Kirchhoff (18241887) In heat transfer, Kirchhoff's law of thermal radiation refers to wavelength-specific radiative emission and absorption by a material body in thermodynamic equilibrium, including radiative exchange equilibrium. A body at temperature radiates electromagnetic energy. A perfect black body in thermodynamic equilibrium absorbs all light that strikes it, and radiates energy according to a unique law of radiative emissive power for temperature , universal for all perfect black bodies. Kirchhoff's law states that: Here, the dimensionless coefficient of absorption (or the absorptivity) is the fraction of incident light (power) that is absorbed by the body when it is radiating and absorbing in thermodynamic equilibrium.
|
|
Artificial light sources are usually evaluated in terms of luminous efficacy of the source, also sometimes called wall-plug efficacy. This is the ratio between the total luminous flux emitted by a device and the total amount of input power (electrical, etc.) it consumes. The luminous efficacy of the source is a measure of the efficiency of the device with the output adjusted to account for the spectral response curve (the luminosity function). When expressed in dimensionless form (for example, as a fraction of the maximum possible luminous efficacy), this value may be called luminous efficiency of a source, overall luminous efficiency or lighting efficiency.
|
|
A scalar field such as temperature or pressure, where intensity of the field is represented by different hues of colors. In mathematics and physics, a scalar field associates a scalar value to every point in a space – possibly physical space. The scalar may either be a (dimensionless) mathematical number or a physical quantity. In a physical context, scalar fields are required to be independent of the choice of reference frame, meaning that any two observers using the same units will agree on the value of the scalar field at the same absolute point in space (or spacetime) regardless of their respective points of origin.
|
|
Often cast in minor roles in larger budget films, Buckman appeared in the Burt Reynolds vehicles Hooper (1978) and The Cannonball Run (1981). Buckman was generally offered largely dimensionless and decorative parts, such as a rollercoaster attendant in the disaster film Rollercoaster (1977), and "Jill - Lamborghini Girl #2" (alongside Adrienne Barbeau) in The Cannonball Run. Buckman also appeared in several 'B' movies from the late 1970s through to the mid-1990s. She appeared in a rape/murder scene, in which her top was torn open and her throat was slit by a man dressed as Santa Claus in the 1984 controversial horror film Silent Night, Deadly Night.
|
|
CGS-emu (or "electromagnetic cgs") units are one of several systems of electromagnetic units within the centimetre gram second system of units; others include CGS- esu, Gaussian units, and Lorentz–Heaviside units. In these other systems, the abcoulomb is not used; CGS-esu and Gaussian units use the statcoulomb is instead, while the Lorentz-Heaviside unit of charge has no specific name. In the electromagnetic cgs system, electric current is a fundamental quantity defined via Ampère's law and takes the permeability as a dimensionless quantity (relative permeability) whose value in a vacuum is unity. As a consequence, the square of the speed of light appears explicitly in some of the equations interrelating quantities in this system.
|
|
In many cases, the manipulated variable output by the PID controller is a dimensionless fraction between 0 and 100% of some maximum possible value, and the translation into real units (such as pumping rate or watts of heater power) is outside the PID controller. The process variable, however, is in dimensioned units such as temperature. It is common in this case to express the gain K_p not as "output per degree", but rather in the reciprocal form of a proportional band 100/K_p, which is "degrees per full output": the range over which the output changes from 0 to 1 (0% to 100%). Beyond this range, the output is saturated, full-off or full-on.
|
|
Net tonnage is calculated by measuring a ship's internal volume and applying mathematical formulae. Net tonnage (often abbreviated as NT, N.T. or nt) is a dimensionless index calculated from the total moulded volume of the ship's cargo spaces by using a mathematical formula. Defined in The International Convention on Tonnage Measurement of Ships that was adopted by the International Maritime Organization in 1969, the net tonnage replaced the earlier net register tonnage (NRT) which denoted the volume of the ship's revenue-earning spaces in "register tons", units of volume equal to . Net tonnage is used to calculate the port duties and should not be taken as less than 30 per cent of the ship's gross tonnage.
|
|
Protocols for centrifugation typically specify the amount of acceleration to be applied to the sample, rather than specifying a rotational speed such as revolutions per minute. This distinction is important because two rotors with different diameters running at the same rotational speed will subject samples to different accelerations. During circular motion the acceleration is the product of the radius and the square of the angular velocity \omega, and the acceleration relative to "g" is traditionally named "relative centrifugal force" (RCF). The acceleration is measured in multiples of "g" (or × "g"), the standard acceleration due to gravity at the Earth's surface, a dimensionless quantity given by the expression:A 19th-century hand cranked laboratory centrifuge.
|
|
Antenna directivity is the ratio of maximum radiation intensity (power per unit surface) radiated by the antenna in the maximum direction divided by the intensity radiated by a hypothetical isotropic antenna radiating the same total power as that antenna. For example, a hypothetical antenna which had a radiated pattern of a hemisphere (1/2 sphere) would have a directivity of 2. Directivity is a dimensionless ratio and may be expressed numerically or in decibels (dB). Directivity is identical to the peak value of the directive gain; these values are specified without respect to antenna efficiency thus differing from the power gain (or simply "gain") whose value is reduced by an antenna's efficiency.
|
|
Being a sensing by the plant of the relative configuration of its parts, it has been called proprioception. This dual sensing and control by gravisensing and proprioception has been formalized into a unifying mathematical model simulating the complete driving of the gravitropic movement. This model has been validated on 11 species sampling the phylogeny of land angiosperms, and on organs of very contrasted sizes, ranging from the small germination of wheat (coleoptile) to the trunk of poplar trees. This model also shows that the entire gravitropic dynamics is controlled by a single dimensionless number called the "Balance Number", and defined as the ratio between the sensitivity to the inclination angle versus gravity and the proprioceptive sensitivity.
|
|
In hydrogeology, volumetric specific storage is much more commonly encountered than mass specific storage. Consequently, the term specific storage generally refers to volumetric specific storage. In terms of measurable physical properties, specific storage can be expressed as :S_s = \gamma_w (\beta_p + n \cdot \beta_w) where :\gamma_w is the specific weight of water (N•m−3 or [ML−2T−2]) :n is the porosity of the material (dimensionless ratio between 0 and 1) :\beta_p is the compressibility of the bulk aquifer material (m2N−1 or [LM−1T2]), and :\beta_w is the compressibility of water (m2N−1 or [LM−1T2]) The compressibility terms relate a given change in stress to a change in volume (a strain).
|
|
In optics, absorbance or decadic absorbance is the common logarithm of the ratio of incident to transmitted radiant power through a material, and spectral absorbance or spectral decadic absorbance is the common logarithm of the ratio of incident to transmitted spectral radiant power through a material. Absorbance is dimensionless, and in particular is not a length, though it is a monotonically increasing function of path length, and approaches zero as the path length approaches zero. The use of the term "optical density" for absorbance is discouraged. In physics, a closely related quantity called "optical depth" is used instead of absorbance: the natural logarithm of the ratio of incident to transmitted radiant power through a material.
|
|
The units and notation above are used when dealing with the physics of mass spectrometry; however, the m/z notation is used for the independent variable in a mass spectrum. This notation eases data interpretation since it is numerically more related to the unified atomic mass unit. For example, if an ion carries one charge the m/z is numerically equivalent to the molecular or atomic mass of the ion in unified atomic mass units (u), where the numerical value of m/Q is abstruse. The m refers to the molecular or atomic mass number and z to the charge number of the ion; however, the quantity of m/z is dimensionless by definition.
|
|
Most notably in 1955 he published an article which described a dimensionless parameter (α) which characterised the nature of unsteady flow; subsequently this has been called the Womersley number. In July 1955, as planned, he moved to WADC to take a post as acting chief of the Analysis Section, System Dynamics Branch Aeronautical Research Laboratory. In 1956, he was promoted to Supervisory Mathematician and then Supervisory Aeronautical Research Engineer (Flight Systems), although he continued to publish on mathematical aspects of blood flow until his early death in 1958. His 1957 monograph on 'An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries' is widely regarded as a major influence in the field.
|
|
Scale analysis is very useful and widely used tool for solving problems in the area of heat transfer and fluid mechanics, pressure-driven wall jet, separating flows behind backward- facing steps, jet diffusion flames, study of linear and non-linear dynamics. Scale analysis is recommended as the premier method for obtaining the most information per unit of intellectual effort, despite the fact that it is a precondition for good analysis in dimensionless form. The object of scale analysis is to use the basic principles of convective heat transfer to produce order-of-magnitude estimates for the quantities of interest. Scale analysis anticipates within a factor of order one when done properly, the expensive results produced by exact analyses.
|
|
Traditional mesh analysis and nodal analysis in electronics lead to a system of linear equations that can be described with a matrix. The behaviour of many electronic components can be described using matrices. Let A be a 2-dimensional vector with the component's input voltage v and input current i as its elements, and let B be a 2-dimensional vector with the component's output voltage v and output current i as its elements. Then the behaviour of the electronic component can be described by B = H · A, where H is a 2 x 2 matrix containing one impedance element (h), one admittance element (h), and two dimensionless elements (h and h).
|
|
The preferred methods in 2019 are measurements of electron anomalous magnetic moments and of photon recoil in atom interferometry. The theory of QED predicts a relationship between the dimensionless magnetic moment of the electron and the fine-structure constant (the magnetic moment of the electron is also referred to as "Landé -factor" and symbolized as ). The most precise value of obtained experimentally (as of 2012) is based on a measurement of using a one-electron so-called "quantum cyclotron" apparatus, together with a calculation via the theory of QED that involved tenth-order Feynman diagrams: : This measurement of has a relative standard uncertainty of . This value and uncertainty are about the same as the latest experimental results.
|
|
A standard linear solid Q model (SLS) for attenuation and dispersion is one of many mathematical Q models that gives a definition of how the earth responds to seismic waves. When a plane wave propagates through a homogeneous viscoelastic medium, the effects of amplitude attenuation and velocity dispersion may be combined conveniently into a single dimensionless parameter, Q, the medium-quality factor. Transmission losses may occur due to friction or fluid movement, and whatever the physical mechanism, they can be conveniently described with an empirical formulation where elastic moduli and propagation velocity are complex functions of frequency. Ursin and ToverudUrsin B. and Toverud T. 2002 Comparison of seismic dispersion and attenuation models.
|
|
The value of a dimensional physical quantity Z is written as the product of a unit [Z] within the dimension and a dimensionless numerical factor, n.For a review of the different conventions in use see: :Z = n \times [Z] = n [Z] When like- dimensioned quantities are added or subtracted or compared, it is convenient to express them in consistent units so that the numerical values of these quantities may be directly added or subtracted. But, in concept, 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 one cannot derive that length by simply adding 1 and 1.
|
|
See also: Bipedalism, Walking, and Gait analysis Gravity has a large influence on walking speed, muscle activity patterns, gait transitions and the mechanics of locomotion, which means that the kinematics of locomotion in space need to be studied in order to optimize movements in that environment. On Earth, the dynamic similarity hypothesis is used to compare gaits between people of different heights and weights. This hypothesis states that different mammals move in a dynamically similar manner when traveling at a speed where they have the same ratio of inertial forces to gravitational forces. This ratio is called the Froude number and is a dimensionless parameter that allows the gait of different sizes and species of animals to be compared.
|
|
The scale of dBZ values can be seen along the bottom of the image. dBZ stands for decibel relative to Z. It is a logarithmic dimensionless technical unit used in radar, mostly in weather radar, to compare the equivalent reflectivity factor (Z) of a remote object (in mm6 per m3) to the return of a droplet of rain with a diameter of 1 mm (1 mm6 per m3). It is proportional to the number of drops per unit volume and the sixth power of drops' diameter and is thus used to estimate the rain or snow intensity. With other variables analyzed from the radar returns it helps to determine the type of precipitation.
|
|
The Hammond–Leffler postulate states that the structure of the transition state more closely resembles either the products or the starting material, depending on which is higher in enthalpy. A transition state that resembles the reactants more than the products is said to be early, while a transition state that resembles the products more than the reactants is said to be late. Thus, the Hammond–Leffler Postulate predicts a late transition state for an endothermic reaction and an early transition state for an exothermic reaction. A dimensionless reaction coordinate that quantifies the lateness of a transition state can be used to test the validity of the Hammond–Leffler postulate for a particular reaction.
|
|
Further, non-dimensionalized Navier–Stokes equations can be beneficial if one is posed with similar physical situations – that is problems where the only changes are those of the basic dimensions of the system. Scaling of Navier–Stokes equation refers to the process of selecting the proper spatial scales – for a certain type of flow – to be used in the non-dimensionalization of the equation. Since the resulting equations need to be dimensionless, a suitable combination of parameters and constants of the equations and flow (domain) characteristics have to be found. As a result of this combination, the number of parameters to be analyzed is reduced and the results may be obtained in terms of the scaled variables.
|
|
In respiratory physiology, specific ventilation is defined as the ratio of the volume of gas entering a region of the lung (ΔV) following an inspiration, divided by the end-expiratory volume (V0) of that same lung region: SV = It is a dimensionless quantity. For the whole human lung, given an indicative tidal volume of 0.6 L and a functional residual capacity of 2.5 L, average SV is of the order of 0.24. The distribution of specific ventilation within the lung can be inferred using Multiple Breath Washout (MBW) experiments Lewis et al, S. M. Lewis, J. W. Evans, and A. A. Jalowayski. Continuous distributions of specific ventilation recovered from inert gas washout, Journal of Applied Physiology, 1978 vol.
|
|
Gross and net register tonnages were replaced by gross tonnage and net tonnage, respectively, when the International Maritime Organization (IMO) adopted The International Convention on Tonnage Measurement of Ships on 23 June 1969. The new tonnage regulations entered into force for all new ships on 18 July 1982, but existing vessels were given a migration period of 12 years to ensure that ships were given reasonable economic safeguards, since port and other dues are charged according to ship's tonnage. Since 18 July 1994 the gross and net tonnages, dimensionless indices calculated from the total moulded volume of the ship and its cargo spaces by mathematical formulae, have been the only official measures of the ship's tonnage.International Convention on Tonnage Measurement of Ships.
|
|
The Showalter index is a dimensionless number computed by taking the temperature at the 850 hPa level which is then taken dry adiabatically up to saturation, then up to the 500 hPa level, which is then subtracted by the observed 500 hPa level temperature. If the value is negative, then the lower portion of the atmosphere is unstable, with thunderstorms expected when the value is below −3. The application of the Showalter index is especially helpful when there is a cool, shallow air mass below 850 hPa that conceals the potential convective lifting. However, the index will underestimate the potential convective lifting if there are cool layers that extend above 850 hPa and it does not consider diurnal radiative changes or moisture below 850 hPa.
|
|
This means that if telephone channels are squeezed in side-by-side into the frequency spectrum, there will be crosstalk from adjacent channels in any given channel. What is required is a much more sophisticated filter that has a flat frequency response in the required passband like a low-Q resonant circuit, but that rapidly falls in response (much faster than 6 dB/octave) at the transition from passband to stopband like a high-Q resonant circuit.Q factor is a dimensionless quantity enumerating the quality of a resonating circuit. It is roughly proportional to the number of oscillations, which a resonator would support after a single external excitation (for example, how many times a guitar string would wobble if pulled).
|
|
Monin–Obukhov (M–O) similarity theory describes non-dimensionalized mean flow and mean temperature in the surface layer under non-neutral conditions as a function of the dimensionless height parameter, named after Russian scientists A. S. Monin and A. M. Obukhov. Similarity theory is an empirical method which describes universal relationships between non-dimensionalized variables of fluids based on the Buckingham Pi theorem. Similarity theory is extensively used in boundary layer meteorology, since relations in turbulent processes are not always resolvable from first principles. An idealized vertical profile of the mean flow for a neutral boundary layer is the logarithmic wind profile derived from Prandtl's mixing length theory, which states that the horizontal component of mean flow is proportional to the logarithm of height.
|
|
Mathematical Q models provide a model of the earth's response to seismic waves. In reflection seismology, the anelastic attenuation factor, often expressed as seismic quality factor or Q, which is inversely proportional to attenuation factor, quantifies the effects of anelastic attenuation on the seismic wavelet caused by fluid movement and grain boundary friction. When a plane wave propagates through a homogeneous viscoelastic medium, the effects of amplitude attenuation and velocity dispersion may be combined conveniently into the single dimensionless parameter, Q. As a seismic wave propagates through a medium, the elastic energy associated with the wave is gradually absorbed by the medium, eventually ending up as heat energy. This is known as absorption (or anelastic attenuation) and will eventually cause the total disappearance of the seismic wave.
|
|
Depending on the field of physics, it may be advantageous to choose one or another extended set of dimensional symbols. In electromagnetism, for example, it may be useful to use dimensions of M, L, T, and Q, where Q represents the dimension of electric charge. In thermodynamics, the base set of dimensions is often extended to include a dimension for temperature, Θ. In chemistry, the amount of substance (the number of molecules divided by the Avogadro constant, ≈ ) is defined as a base dimension, N, as well. In the interaction of relativistic plasma with strong laser pulses, a dimensionless relativistic similarity parameter, connected with the symmetry properties of the collisionless Vlasov equation, is constructed from the plasma-, electron- and critical-densities in addition to the electromagnetic vector potential.
|
|
Above , cellulose was observed to exhibit transition boiling with violent bubbling and associated reduction in heat transfer. Liftoff of the cellulose droplet (depicted at the right) was observed to occur above about associated with a dramatic reduction in heat transfer. High speed photography of the reactive Leidenfrost effect of cellulose on porous surfaces (macroporous alumina) was also shown to suppress the reactive Leidenfrost effect and enhance overall heat transfer rates to the particle from the surface. The new phenomenon of a 'reactive Leidenfrost (RL) effect' was characterized by a dimensionless quantity (φRL= τconv/τrxn), which relates the time constant of solid particle heat transfer to the time constant of particle reaction, with the reactive Leidenfrost effect occurring for 10−1< φRL< 10+1.
|
|
In 1717 Isaac Newton speculated that light particles and matter particles were interconvertible in "Query 30" of the Opticks, where he asks: "Are not the gross bodies and light convertible into one another, and may not bodies receive much of their activity from the particles of light which enter their composition?" In 1734 the Swedish scientist and theologian Emanuel Swedenborg in his Principia theorized that all matter is ultimately composed of dimensionless points of "pure and total motion". He described this motion as being without force, direction or speed, but having the potential for force, direction and speed everywhere within it. During the nineteenth century there were several speculative attempts to show that mass and energy were proportional in various ether theories.
|
|
For this reason, urea is analysed in patients undergoing dialysis as the adequacy of the treatment can be assessed by the dimensionless parameter Kt/V , which can be calculated from the concentration of urea in the blood. Urea has also been studied as an excipient in Drug-coated Balloon (DCB) coating formulation to enhance local drug delivery to stenotic blood vessels. Urea, when used as an excipient in small doses (~3μg/mm2) to coat DCB surface was found to form crystals that increase drug transfer without adverse toxic effects on vascular endothelial cells. Urea labeled with carbon-14 or carbon-13 is used in the urea breath test, which is used to detect the presence of the bacterium Helicobacter pylori (H.
|
|
For instance, Hermann Hankel has written of the works of Diophantus that "not the slightest trace of a general, comprehensive method is discernible; each problem calls for some special method which refuses to work even for the most closely related problems". In contrast, the thesis of Bashmakova's book is that Diophantus indeed had general methods, which can be inferred from the surviving record of his solutions to these problems. The opening chapter of the books tells what is known of Diophantus and his contemporaries, and surveys the problems published by Diophantus. The second chapter reviews the mathematics known to Diophantus, including his development of negative numbers, rational numbers, and powers of numbers, and his philosophy of mathematics treating numbers as dimensionless quantities, a necessary preliminary to the use of inhomogeneous polynomials.
|
|
Also introduced by Chou in 1976 was the median-effect plot which is a plot of log (D) vs log [(fa)/(1-fa)] or log [(fa)/(fu)] yields a straight line with slope (m) and the x-intercept of log (Dm), where Dm equals to the anti-log of the x-intercept. This unique theory holds true for all dose-effect curves that follows the physico-chemical principle of the mass-action law, for all entities regardless of the first- order or higher-order dynamics, and regardless of unit or mechanism of actions.[3] MEE is derived by system analysis using enzyme kinetics and mathematical inductions and deductions where hundreds of mechanism specific individual equations are reduced to a single general equation. [3] Both left and right sides of the MEE are dimensionless quantities.
|
|
In aerodynamics, the zero-lift drag coefficient C_{D,0} is a dimensionless parameter which relates an aircraft's zero-lift drag force to its size, speed, and flying altitude. Mathematically, zero-lift drag coefficient is defined as C_{D,0} = C_D - C_{D,i}, where C_D is the total drag coefficient for a given power, speed, and altitude, and C_{D,i} is the lift-induced drag coefficient at the same conditions. Thus, zero-lift drag coefficient is reflective of parasitic drag which makes it very useful in understanding how "clean" or streamlined an aircraft's aerodynamics are. For example, a Sopwith Camel biplane of World War I which had many wires and bracing struts as well as fixed landing gear, had a zero-lift drag coefficient of approximately 0.0378.
|
|
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 contrasted with a mathematical constant, which has a fixed numerical value, but does not directly involve any physical measurement. There are many physical constants in science, some of the most widely recognized being the speed of light in vacuum c, the gravitational constant G, the Planck constant h, the electric constant ε0, and the elementary charge e. Physical constants can take many dimensional forms: the speed of light signifies a maximum speed for any object and its dimension is length divided by time; while the fine-structure constant α, which characterizes the strength of the electromagnetic interaction, is dimensionless.
|
|
Some physicists have explored the notion that if the dimensionless physical constants had sufficiently different values, our Universe would be so radically different that intelligent life would probably not have emerged, and that our Universe therefore seems to be fine-tuned for intelligent life. However, the phase space of the possible constants and their values is unknowable, so any conclusions drawn from such arguments are unsupported. The anthropic principle states a logical truism: the fact of our existence as intelligent beings who can measure physical constants requires those constants to be such that beings like us can exist. There are a variety of interpretations of the constants' values, including that of a divine creator (the apparent fine-tuning is actual and intentional), or that ours is one universe of many in a multiverse (e.g.
|
|
Rocket vehicle Thrust-to-weight ratio vs specific impulse for different propellant technologies The thrust-to-weight ratio of a rocket, or rocket-propelled vehicle, is an indicator of its acceleration expressed in multiples of gravitational acceleration g.George P. Sutton & Oscar Biblarz, Rocket Propulsion Elements (p. 442, 7th edition) "thrust-to- weight ratio F/Wg is a dimensionless parameter that is identical to the acceleration of the rocket propulsion system (expressed in multiples of g0) if it could fly by itself in a gravity-free vacuum" Rockets and rocket-propelled vehicles operate in a wide range of gravitational environments, including the weightless environment. The thrust-to-weight ratio is usually calculated from initial gross weight at sea level on earthGeorge P. Sutton & Oscar Biblarz, Rocket Propulsion Elements (p.
|
|
Figure 3: M-y Curves for Various Unit Flowrates M-y diagrams can provide information about the characteristics and behavior of a certain discharge in a channel. Primarily, an M-y diagram will show which flow depths correspond to supercritical or subcritical flow for a given discharge, as well as defining the critical depth and critical momentum of a flow. In addition, M-y diagrams can aid in finding conjugate depths of flow that have the same specific force or momentum function, as in the case of flow depths on either side of a hydraulic jump. A dimensionless form of the M-y diagram representing any unit discharge can be created and utilized in place of the particular M-y curves discussed here and referred to in Figure 3.
|
|
Developments in physics may lead to either a reduction or an extension of their number: discovery of new particles, or new relationships between physical phenomena, would introduce new constants, while the development of a more fundamental theory might allow the derivation of several constants from a more fundamental constant. A long- sought goal of theoretical physics is to find first principles ("Theory of Everything") from which all of the fundamental dimensionless constants can be calculated and compared to the measured values. The large number of fundamental constants required in the Standard Model has been regarded as unsatisfactory since the theory's formulation in the 1970s. The desire for a theory that would allow the calculation of particle masses is a core motivation for the search for "Physics beyond the Standard Model".
|
|
This hypothetical unaffected observer on the outside might observe that light now propagates at half the speed that it previously did (as well as all other observed velocities) but it would still travel of our new metres in the time elapsed by one of our new seconds (c × 4 ÷ 2 continues to equal ). We would not notice any difference. This contradicts what George Gamow writes in his book Mr. Tompkins; there, Gamow suggests that if a dimension-dependent universal constant such as c changed significantly, we would easily notice the difference. The disagreement is better thought of as the ambiguity in the phrase "changing a physical constant"; what would happen depends on whether (1) all other dimensionless constants were kept the same, or whether (2) all other dimension-dependent constants are kept the same.
|
|
Relative isotopic mass (a property of a single atom) is not to be confused with the averaged quantity atomic weight (see above), that is an average of values for many atoms in a given sample of a chemical element. While atomic mass is an absolute mass, relative isotopic mass is a dimensionless number with no units. This loss of units results from the use of a scaling ratio with respect to a carbon-12 standard, and the word "relative" in the term "relative isotopic mass" refers to this scaling relative to carbon-12. The relative isotopic mass, then, is the mass of a given isotope (specifically, any single nuclide), when this value is scaled by the mass of carbon-12, where the latter has to be determined experimentally.
|
|
In order to be able to use the available standard indexes to locations where index data is not available we have to incorporate a new term called the Location Factor (LF) to the standard index value. It is a dimensionless value for a particular location relative to either of the above- mentioned basis. Cost in A = Cost in USGC x LF(A) where A is the location for which cost is being evaluated and LF(A) is the location factor for the location A relative to USGC Location factors are greatly influenced by currency exchange rates due to their significant effect on Index value and hence vary drastically with time. Over the past couple of decades the location factors for various locations are trending close to the value 1.
|
|
In physics, optical depth or optical thickness is the natural logarithm of the ratio of incident to transmitted radiant power through a material, and spectral optical depth or spectral optical thickness is the natural logarithm of the ratio of incident to transmitted spectral radiant power through a material. Optical depth is dimensionless, and in particular is not a length, though it is a monotonically increasing function of optical path length, and approaches zero as the path length approaches zero. The use of the term "optical density" for optical depth is discouraged. In chemistry, a closely related quantity called "absorbance" or "decadic absorbance" is used instead of optical depth: the common logarithm of the ratio of incident to transmitted radiant power through a material, that is the optical depth divided by ln 10.
|
|
While references to comparing apples and oranges are often a rhetorical device, references to adding apples and oranges are made in the case of teaching students the proper uses of units. Here, the admonition not to "add apples and oranges" refers to the requirement that two quantities with different units may not be combined by addition, although they may always be combined in ratio form by multiplication, so that multiplying ratios of apples and oranges is allowed. Similarly, the concept of this distinction is often used metaphorically in elementary algebra. The admonition is really more of a mnemonic, since in general counts of objects have no intrinsic unit and, for example, a number count of apples may be dimensionless or have dimension fruit; in either of these two cases, apples and oranges may indeed be added.
|
|
If the universe were 10 times older than it actually is, most stars would be too old to remain on the main sequence and would have turned into white dwarfs, aside from the dimmest red dwarfs, and stable planetary systems would have already come to an end. Thus, Dicke explained the coincidence between large dimensionless numbers constructed from the constants of physics and the age of the universe, a coincidence that inspired Dirac's varying-G theory. Dicke later reasoned that the density of matter in the universe must be almost exactly the critical density needed to prevent the Big Crunch (the "Dicke coincidences" argument). The most recent measurements may suggest that the observed density of baryonic matter, and some theoretical predictions of the amount of dark matter account for about 30% of this critical density, with the rest contributed by a cosmological constant.
|
|
Computational fluid dynamics (CFD) techniques are the standard tools to analyse and evaluate heat exchangers and similar equipment. However, for quick calculation purposes, the evaluation of DSSHEs are usually carried out with the help of ad hoc (semi)empirical correlations based on the Buckingham π theorem: :Fa = Fa(Re, Re', n, ...) for pressure loss and :Nu = Nu(Re, Re', Pr, Fa, L/D, N, ...) for heat transfer, where Nu is the Nusselt number, Re is the standard Reynolds number based on the inner diameter of the tube, Re' is the specific Reynolds number based on the wiping frequency, Pr is the Prandtl number, Fa is the Fanning friction factor, L is the length of the tube, D is the inner diameter of the tube, n is the number of blades and the dots account for any other relevant dimensionless parameters.
|
|
This dimensionless coefficient will be a combination of geometric factors such as , the Reynolds number and (outside the laminar regime) the relative roughness of the pipe (the ratio of the roughness height to the hydraulic diameter). Note that the dynamic pressure is not the kinetic energy of the fluid per unit volume, for the following reasons. Even in the case of laminar flow, where all the flow lines are parallel to the length of the pipe, the velocity of the fluid on the inner surface of the pipe is zero due to viscosity, and the velocity in the center of the pipe must therefore be larger than the average velocity obtained by dividing the volumetric flow rate by the wet area. The average kinetic energy then involves the root mean-square velocity, which always exceeds the mean velocity.
|
|
In the models of LDM, cracking or local buckling as well as plasticity are lumped at the inelastic hinges. As in continuum damage mechanics, LDM uses state variables to represent the effects of damage on the remaining stiffness and strength of the frame structure. In reinforced concrete structures, the damage state variable quantifies the crack density in the plastic hinge zone; in unreinforced concrete components and steel beams, it is a dimensionless measure of the crack surface;Amorim, D.L.N.D.F., Proença, S.P.B.,Flórez-López, J. “Simplified modeling of cracking in concrete: Application in tunnel linings” Engineering Structures, 70, pp. 23-25 (2014) in tubular steel elements, the damage variable measures the degree of local bucklingMarante, M.E., Picón, R., Guerrero, N. And Flórez-López, J. “Local buckling in tridimensional frames: experimentation and simplified analysis” 9(2012) 691 – 712 Latin-American Journal of Solids and Structures.
|
|
A schematic showing the relationship between dBu (the voltage source) and dBm (the power dissipated as heat by the 600 Ω resistor) dBm (sometimes dBmW or decibel-milliwatts) is a unit of level used to indicate that a power ratio is expressed in decibels (dB) with reference to one milliwatt (mW). It is used in radio, microwave and fiber-optical communication networks as a convenient measure of absolute power because of its capability to express both very large and very small values in a short form compared to dBW, which is referenced to one watt (1000 mW). Since it is referenced to the watt, it is an absolute unit, used when measuring absolute power. By comparison, the decibel (dB) is a dimensionless unit, used for quantifying the ratio between two values, such as signal-to-noise ratio.
|
|
Victor Szebehely Victor G. Szebehely (August 21, 1921 – September 13, 1997) was a key figure in the development and success of the Apollo program. In 1956, a dimensionless number used in time-dependent unsteady flows was named "Szebehely's number," (In the September and October 1977 issues of the journal Celestial Mechanics, volume 16, an equation used to determine the gravitational potential of the Earth, planets, satellites, and galaxies was named "Szebehely's equation". He worked with General Electric, Yale University, the Royal Netherlands Navy, the United States Air Force, NASA, and the University of Texas at Austin. One of his areas of research was orbital debris and planetary defense against meteor impacts His first book, The Theory of Orbits, is an important work in orbital mechanics, being the definitive text on the restricted three-body problem as applicable to an Earth-Moon spacecraft system such as Apollo.
|
|
Overall, the total irradiation generated by the sun, and received by the earth is a major factor affecting Earth climate. The total solar irradiation received by Earth's surface can be calculated mathematically. This can be done through the following equation: Gt=GND+Gd+GR where GND is direct irradiation from the sun, Gd is diffuse irradiation from nearby surroundings being heated up by the sun, and GR is the reflected irradiation from other nearby surfaces. If all that is to be considered is the flat surface of the Earth, then the equation turns into the following: Gt=GND (sin β+C) where β is the solar altitude angle, which is basically the angle the sun appears to be above the horizon from wherever the solar irradiation is to be calculated, and C is a dimensionless ratio equal to anywhere between 0.103 and 0.138 depending on what day of the year it is where the irradiation is to be calculated.
|
|
In fluid dynamics, the drag equation is a formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid. The equation is: ;F_D\, =\, \tfrac12\, \rho\, u^2\, C_D\, A :F_D is the drag force, which is by definition the force component in the direction of the flow velocity, :\rho is the mass density of the fluid,Note that for the Earth's atmosphere, the air density can be found using the barometric formula. Air is 1.293 kg/m3 at 0°C and 1 atmosphere :u is the flow velocity relative to the object, :A is the reference area, and :C_D is the drag coefficient – a dimensionless coefficient related to the object's geometry and taking into account both skin friction and form drag. If the fluid is a liquid, C_D depends on the Reynolds number; if the fluid is a gas, C_D depends on both the Reynolds number and the Mach number.
|
|
Bjorken discovered in 1968 what is known as light-cone scaling (or Bjorken scaling), a phenomenon in the deep inelastic scattering of light on strongly interacting particles, known as hadrons (such as protons and neutrons): Experimentally observed hadrons behave as collections of virtually independent point-like constituents when probed at high energies. Properties of these hadrons scale, that is, they are determined not by the absolute energy of an experiment, but, instead, by dimensionless kinematic quantities, such as a scattering angle or the ratio of the energy to a momentum transfer. Because increasing energy implies potentially improved spatial resolution, scaling implies independence of the absolute resolution scale, and hence effectively point-like substructure. This observation was critical to the recognition of quarks as actual elementary particles (rather than just convenient theoretical constructs), and led to the theory of strong interactions known as quantum chromodynamics, where it was understood in terms of the asymptotic freedom property.
|
|
The measure of a harmonic oscillator's resistance to disturbances to its oscillation period is a dimensionless parameter called the Q factor equal to the resonant frequency divided by the resonance width. has an excellent comprehensive discussion of the controversy over the applicability of Q to the accuracy of pendulums. The higher the Q, the smaller the resonance width, and the more constant the frequency or period of the oscillator for a given disturbance. The reciprocal of the Q is roughly proportional to the limiting accuracy achievable by a harmonic oscillator as a time standard.Matthys, 2004, p.32, fig. 7.2 and text The Q is related to how long it takes for the oscillations of an oscillator to die out. The Q of a pendulum can be measured by counting the number of oscillations it takes for the amplitude of the pendulum's swing to decay to 1/e = 36.8% of its initial swing, and multiplying by 2π.
|
|
In the pancake world, this would manifest if the creatures were living on an enormous sphere rather than on a plane. In this case, when they wander around their sphere, they would eventually come to realize that translations are not entirely separate from rotations, because if they move around on the surface of a sphere, when they come back to where they started, they find that they have been rotated by the holonomy of parallel transport on the sphere. If the universe is the same everywhere (homogeneous) and there are no preferred directions (isotropic), then there are not many options for the symmetry group: they either live on a flat plane, or on a sphere with a constant positive curvature, or on a Lobachevski plane with constant negative curvature. If they are not living on the plane, they can describe positions using dimensionless angles, the same parameters that describe rotations, so that translations and rotations are nominally unified.
|
|
The fact that we have assigned incompatible dimensions to Length, Time and Mass is, according to this point of view, just a matter of convention, borne out of the fact that before the advent of modern physics, there was no way to relate mass, length, and time to each other. The three independent dimensionful constants: c, ħ, and G, in the fundamental equations of physics must then be seen as mere conversion factors to convert Mass, Time and Length into each other. Just as in the case of critical properties of lattice models, one can recover the results of dimensional analysis in the appropriate scaling limit; e.g., dimensional analysis in mechanics can be derived by reinserting the constants ħ, c, and G (but we can now consider them to be dimensionless) and demanding that a nonsingular relation between quantities exists in the limit c\rightarrow \infty, \hbar\rightarrow 0 and G\rightarrow 0.
|
|
Similarly, for a point mass m the moment of inertia is defined as, :I=r^2m :where r is the radius of the point mass from the center of rotation, and for any collection of particles m_i as the sum, :\sum_i I_i = \sum_i r_i^2m_i Angular momentum's dependence on position and shape is reflected in its units versus linear momentum: kg⋅m2/s, N⋅m⋅s, or J⋅s for angular momentum versus kg⋅m/s or N⋅s for linear momentum. When calculating angular momentum as the product of the moment of inertia times the angular velocity, the angular velocity must be expressed in radians per second, where the radian assumes the dimensionless value of unity. (When performing dimensional analysis, it may be productive to use orientational analysis which treats radians as a base unit, but this is outside the scope of the International system of units). Angular momentum's units can be interpreted as torque⋅time or as energy⋅time per angle.
|
|
In slightly different terms, the emissive power of an arbitrary opaque body of fixed size and shape at a definite temperature can be described by a dimensionless ratio, sometimes called the emissivity: the ratio of the emissive power of the body to the emissive power of a black body of the same size and shape at the same fixed temperature. With this definition, Kirchhoff's law states, in simpler language: In some cases, emissive power and absorptivity may be defined to depend on angle, as described below. The condition of thermodynamic equilibrium is necessary in the statement, because the equality of emissivity and absorptivity often does not hold when the material of the body is not in thermodynamic equilibrium. Kirchhoff's law has another corollary: the emissivity cannot exceed one (because the absorptivity cannot, by conservation of energy), so it is not possible to thermally radiate more energy than a black body, at equilibrium.
|
|
His ideas were not widely accepted, and subsequent experiments have shown that they were wrong (for example, none of the measurements of the fine-structure constant suggest an integer value; in 2018 it was measured at α = 1/137.035999046(27)). Though his derivations and equations were unfounded, Eddington was the first physicist to recognize the significance of universal dimensionless constants, now considered among the most critical components of major physical theories such as the Standard Model and ΛCDM cosmology.Prialnik, D. K., An Introduction to the Theory of Stellar Structure and Evolution (Cambridge: Cambridge University Press, 2000), p. 82. He was also the first to argue for the importance of the cosmological constant Λ itself, considering it vital for explaining the expansion of the universe, at a time when most physicists (including its discoverer, Albert Einstein) considered it an outright mistake or mathematical artifact and assumed a value of zero: this at least proved prescient, and a significant positive Λ features prominently in ΛCDM.
|
|
In general when the wind blows on a building then the air pressure, Pw varies all over the building surface (Fig 7). Fig 7 Wind Pressure Distribution around a Building (Liddament, 1986) where Poa reference pressure (Pa) Cpwind pressure coefficient (dimensionless) Liddament, and CIBSE, provide approximate wind pressure coefficient data for low rise buildings (up to 3 storeys). For a square plan building on an exposed site with the wind blowing directly on to the face of the building the wind pressure coefficients are as shown in Fig 8. For a wind speed of 5.7 m/s at ridge height (taken as 8m) there is zero pressure difference across the side walls when the building is depressurised to -10 Pa. The insulation in the windward and leeward walls is behaving dynamically in the contra-flux mode with U-values of 0.0008 W/(m2K) and 0.1 W/(m2K) respectively. Since the building has a square footprint the average U-value for the walls is 0.1252 W/m2K.
|
|
Amedeo Avogadro The Avogadro constant (NA or L) is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. Its SI unit is the reciprocal mole, and it is defined as It is named after the Italian scientist Amedeo Avogadro. The numeric value of the Avogadro constant expressed in reciprocal mole, a dimensionless number, is called the Avogadro number, sometimes denoted N or N0, which is thus the number of particles that are contained in one mole, exactly . The value of the Avogadro constant was chosen so that the mass of one mole of a chemical compound, in grams, is numerically equal (for all practical purposes) to the average mass of one molecule of the compound, in daltons (universal atomic mass units); one dalton being of the mass of one carbon-12 atom, which is approximately the mass of one nucleon (proton or neutron).
|
|
The expansion of production, characterised by changes of the accumulated value K, requires additional labour L and substitutive work P, so that dynamics of the production factors can be written as the balance equations The first terms in the right side of these relations describe the increase in the quantities caused by gross investments I accumulated in a material form of production equipment. Equations (1) introduce the quality of investment that is technological characteristics of production equipment: coefficients of labour and energy requirement, which can be manipulated both in the dimensional, \lambda and \varepsilon, and dimensionless (with a bar on the top) forms, \bar\lambda = \lambda K / L and \bar\varepsilon = \varepsilon K / P. The second terms on the right side of equations (1) reflect the decrease in the corresponding quantities due to the removal of a part of the production equipment from service with the depreciation coefficient \mu, which is taken equal for all factors due to assumption, that the technological characteristics of the equipment do not change after its installation, otherwise the dynamic equations takes a more complex form.
|
|
The discrepancy between D and D^2 shows that a balance of energy release and dissipation rate can exist for every size D only if \sigma_N decreases with increasing D. If the energy dissipated within the damage zone of width h is added, one obtains the Bažant (1984) size effect law (Type 2): (Fig. 4c,d) where B, f'_t, D_0 = constants, where f'_t = tensile strength of material, and B accounts for the structure geometry. For more complex geometries such an intuitive derivation is not possible. However, dimensional analysis coupled with asymptotic matching showed that Eq. 8 is applicable in general, and that the dependence of its parameters on the structure geometry has approximately the following form: where c_f \approx half of the FPZ length, \alpha_0 = a/D = relative initial crack length (which is constant for geometrically similar scaling); g(\alpha_0) = k^2(\alpha_0) = dimensionless energy release function of linear elastic fracture mechanics (LEFM), which brings about the effect of structure geometry; k(\alpha_0)= K(\alpha_0) b \sqrt D /P, and K = stress intensity factor.
|
|
In analytic geometry, the isoperimetric ratio of a simple closed curve in the Euclidean plane is the ratio , where is the length of the curve and is its area. It is a dimensionless quantity that is invariant under similarity transformations of the curve. According to the isoperimetric inequality, the isoperimetric ratio has its minimum value, 4, for a circle; any other curve has a larger value.. Thus, the isoperimetric ratio can be used to measure how far from circular a shape is. The curve-shortening flow decreases the isoperimetric ratio of any smooth convex curve so that, in the limit as the curve shrinks to a point, the ratio becomes 4.. For higher-dimensional bodies of dimension d, the isoperimetric ratio can similarly be defined as where B is the surface area of the body (the measure of its boundary) and V is its volume (the measure of its interior).. Other related quantities include the Cheeger constant of a Riemannian manifold and the (differently defined) Cheeger constant of a graph..
|
|
In this study, the authors found that the best estimate of tomorrow's price is not yesterday's price (as the efficient- market hypothesis would indicate), nor is it the pure momentum price (namely, the same relative price change from yesterday to today continues from today to tomorrow). But rather it is almost exactly halfway between the two. Starting from the characterization of the past time evolution of market prices in terms of price velocity and price acceleration, an attempt towards a general framework for technical analysis has been developed, with the goal of establishing a principled classification of the possible patterns characterizing the deviation or defects from the random walk market state and its time translational invariant properties.J. V. Andersen, S. Gluzman and D. Sornette, Fundamental Framework for Technical Analysis, European Physical Journal B 14, 579-601 (2000) The classification relies on two dimensionless parameters, the Froude number characterizing the relative strength of the acceleration with respect to the velocity and the time horizon forecast dimensionalized to the training period.
|
|
Shannon's efforts to find a way to quantify the information contained in, for example, a telegraph message, led him unexpectedly to a formula with the same form as Boltzmann's. In an article in the August 2003 issue of Scientific American titled "Information in the Holographic Universe", Bekenstein summarizes that "Thermodynamic entropy and Shannon entropy are conceptually equivalent: the number of arrangements that are counted by Boltzmann entropy reflects the amount of Shannon information one would need to implement any particular arrangement" of matter and energy. The only salient difference between the thermodynamic entropy of physics and Shannon's entropy of information is in the units of measure; the former is expressed in units of energy divided by temperature, the latter in essentially dimensionless "bits" of information. The holographic principle states that the entropy of ordinary mass (not just black holes) is also proportional to surface area and not volume; that volume itself is illusory and the universe is really a hologram which is isomorphic to the information "inscribed" on the surface of its boundary.
|
|
The SI definition given by the International Committee for Weights and Measures (CIPM) says: "The quantity dose equivalent H is the product of the absorbed dose D of ionizing radiation and the dimensionless factor Q (quality factor) defined as a function of linear energy transfer by the ICRU" :H = Q × D The value of Q is not defined further by CIPM, but it requires the use of the relevant ICRU recommendations to provide this value. The CIPM also says that "in order to avoid any risk of confusion between the absorbed dose D and the dose equivalent H, the special names for the respective units should be used, that is, the name gray should be used instead of joules per kilogram for the unit of absorbed dose D and the name sievert instead of joules per kilogram for the unit of dose equivalent H". In summary: The gray – quantity D - Absorbed dose :1 Gy = 1 joule/kilogram – a physical quantity. 1 Gy is the deposit of a joule of radiation energy per kg of matter or tissue. The sievert – quantity H - Dose equivalent :1 Sv = 1 joule/kilogram – a biological effect.
|
|