In engineering and science, dimensional analysis is the analysis of
the relationships between different physical quantities by identifying
their base quantities (such as length, mass, time, and electric
charge) and units of measure (such as miles vs. kilometers, or pounds
vs. kilograms vs. grams) and tracking these dimensions as calculations
or comparisons are performed. Converting from one dimensional unit to
another is often somewhat complex. Dimensional analysis, or more
specifically the factor-label method, also known as the unit-factor
method, is a widely used technique for such conversions using the
rules of algebra.[1][2][3]
The concept of physical dimension was introduced by
Contents 1 Concrete numbers and base units 1.1 Percentages and derivatives 2 Conversion factor 3 Dimensional homogeneity 4 The factor-label method for converting units 4.1 Checking equations that involve dimensions 4.2 Limitations 5 Applications 5.1 Mathematics 5.2 Finance, economics, and accounting 5.3 Fluid mechanics 6 History 7 Mathematical examples 7.1 Definition 7.2 Mathematical properties 7.3 Mechanics 7.4 Other fields of physics and chemistry 7.5 Polynomials and transcendental functions 7.6 Incorporating units 7.7 Position vs displacement 7.8 Orientation and frame of reference 8 Examples 8.1 A simple example: period of a harmonic oscillator 8.2 A more complex example: energy of a vibrating wire 8.3 A third example: demand versus capacity for a rotating disc 9 Extensions 9.1 Huntley's extension: directed dimensions 9.2 Siano's extension: orientational analysis 10 Dimensionless concepts 10.1 Constants 10.2 Formalisms 11 Dimensional equivalences 11.1 SI units 11.2 Natural units 12 See also 12.1 Related areas of math 13 Notes 14 References 15 External links 15.1 Converting units Concrete numbers and base units[edit] Many parameters and measurements in the physical sciences and engineering are expressed as a concrete number – a numerical quantity and a corresponding dimensional unit. Often a quantity is expressed in terms of several other quantities; for example, speed is a combination of length and time, e.g. 60 miles per hour or 1.4 kilometers per second. Compound relations with "per" are expressed with division, e.g. 60 mi/1 h. Other relations can involve multiplication (often shown with a centered dot or juxtaposition), powers (like m2 for square meters), or combinations thereof. A set of base units for a system of measurement is a conventionally chosen set of units, none of which can be expressed as a combination of the others, and in terms of which all the remaining units of the system can be expressed.[5] For example, units for length and time are normally chosen as base units. Units for volume, however, can be factored into the base units of length (m3), thus they are considered derived or compound units. Sometimes the names of units obscure the fact that they are derived units. For example, a newton (N) is a unit of force, which will have units of mass (kg) times acceleration (m⋅s−2). The newton is defined as 1 N = 1 kg⋅m⋅s−2. Percentages and derivatives[edit] Percentages are dimensionless quantities, since they are ratios of two quantities with the same dimensions. In other words, the % sign can be read as "hundredths", since 1% = 1/100. Taking a derivative with respect to a quantity adds the dimension of the variable one is differentiating with respect to, in the denominator. Thus: position (x) has the dimension L (length); derivative of position with respect to time (dx/dt, velocity) has dimension LT−1 – length from position, time due to the derivative; the second derivative (d2x/dt2 = (dx/dt) / dt, acceleration) has dimension LT−2. In economics, one distinguishes between stocks and flows: a stock has units of "units" (say, widgets or dollars), while a flow is a derivative of a stock, and has units of "units/time" (say, dollars/year). In some contexts, dimensional quantities are expressed as dimensionless quantities or percentages by omitting some dimensions. For example, debt-to-GDP ratios are generally expressed as percentages: total debt outstanding (dimension of currency) divided by annual GDP (dimension of currency) – but one may argue that in comparing a stock to a flow, annual GDP should have dimensions of currency/time (dollars/year, for instance), and thus Debt-to-GDP should have units of years, which indicates that Debt-to-GDP is the number of years needed for a constant GDP to pay the debt, if all GDP is spent on the debt and the debt is otherwise unchanged. Conversion factor[edit] Main article: Conversion factor In dimensional analysis, a ratio which converts one unit of measure into another without changing the quantity is called a conversion factor. For example, kPa and bar are both units of pressure, and 100 kPa = 1 bar. The rules of algebra allow both sides of an equation to be divided by the same expression, so this is equivalent to 100 kPa / 1 bar = 1. Since any quantity can be multiplied by 1 without changing it, the expression "100 kPa / 1 bar" can be used to convert from bars to kPa by multiplying it with the quantity to be converted, including units. For example, 5 bar × 100 kPa / 1 bar = 500 kPa because 5 × 100 / 1 = 500, and bar/bar cancels out, so 5 bar = 500 kPa. Dimensional homogeneity[edit] See also: Apples and oranges The most basic rule of dimensional analysis is that of dimensional homogeneity.[6] Only commensurable quantities (physical quantities having the same dimension) may be compared, equated, added, or subtracted. However, the dimensions form an abelian group under multiplication, so: One may take ratios of incommensurable quantities (quantities with different dimensions), and multiply or divide them. For example, it makes no sense to ask whether 1 hour is more, the
same, or less than 1 kilometer, as these have different dimensions,
nor to add 1 hour to 1 kilometer. However, it makes perfect sense to
ask whether 1 mile is more, the same, or less than 1 kilometer being
the same dimension of physical quantity even though the units are
different. On the other hand, if an object travels 100 km in 2
hours, one may divide these and conclude that the object's average
speed was 50 km/h.
The rule implies that in a physically meaningful expression only
quantities of the same dimension can be added, subtracted, or
compared. For example, if mman, mrat and Lman denote, respectively,
the mass of some man, the mass of a rat and the length of that man,
the dimensionally homogeneous expression mman + mrat is meaningful,
but the heterogeneous expression mman + Lman is meaningless. However,
mman/L2man is fine. 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.
Even when two physical quantities have identical dimensions, it may
nevertheless be meaningless to compare or add them. For example,
although torque and energy share the dimension L2MT−2, they are
fundamentally different physical quantities.
To compare, add, or subtract quantities with the same dimensions but
expressed in different units, the standard procedure is first to
convert them all to the same units. For example, to compare 32 metres
with 35 yards, use 1 yard = 0.9144 m to convert 35 yards to
32.004 m.
A related principle is that any physical law that accurately describes
the real world must be independent of the units used to measure the
physical variables.[7] For example,
10 mile 1 hour × 1609.344 meter 1 mile × 1 hour 3600 second = 4.4704 meter second . displaystyle frac 10 cancel text mile 1 cancel text hour times frac 1609.344 text meter 1 cancel text mile times frac 1 cancel text hour 3600 text second =4.4704 frac text meter text second . It can be seen that each conversion factor is equivalent to the value of one. For example, starting with 1 mile = 1609.344 meters and dividing both sides of the equation by 1 mile yields 1 mile / 1 mile = 1609.344 meters / 1 mile, which when simplified yields 1 = 1609.344 meters / 1 mile. So, when the units mile and hour are cancelled out and the arithmetic is done, 10 miles per hour converts to 4.4704 meters per second. As a more complex example, the concentration of nitrogen oxides (i.e., NO x displaystyle color Blue ce NO _ x ) in the flue gas from an industrial furnace can be converted to a mass flow rate expressed in grams per hour (i.e., g/h) of NO x displaystyle ce NO _ x by using the following information as shown below:
10 m 3
NO x 10 6
m 3
gas × 20 m 3
gas 1 minute × 60 minute 1 hour × 1 mol NO x 22.414 m 3
NO x × 46 kg NO x 1 mol NO x × 1000 g NO x 1 kg NO x = 24.63 g NO x hour displaystyle frac 10 cancel ce m ^ 3 ce NO _ x 10^ 6 cancel ce m ^ 3 ce gas times frac 20 cancel ce m ^ 3 ce gas 1 cancel ce minute times frac 60 cancel ce minute 1 ce hour times frac 1 cancel ce mol NO _ x 22.414 cancel ce m ^ 3 ce NO _ x times frac 46 cancel ce kg NO _ x 1 cancel ce mol NO _ x times frac 1000 ce g NO _ x 1 cancel ce kg NO _ x =24.63 frac ce g NO _ x ce hour After canceling out any dimensional units that appear both in the
numerators and denominators of the fractions in the above equation,
the
the pressure P is in pascals (Pa) the volume V is in cubic meters (m3) the amount of substance n is in moles (mol) the universal gas law constant R is 8.3145 Pa⋅m3/(mol⋅K) the temperature T is in kelvins (K) Pa ⋅ m 3 = mol 1 × Pa ⋅ m 3 mol
K × K 1 displaystyle ce Pa.m^3 = frac cancel ce mol 1 times frac ce Pa.m^3 cancel ce mol cancel ce K times frac cancel ce K 1 As can be seen, when the dimensional units appearing in the numerator and denominator of the equation's right hand side are cancelled out, both sides of the equation have the same dimensional units. Limitations[edit] The factor-label method can convert only unit quantities for which the units are in a linear relationship intersecting at 0. Most units fit this paradigm. An example for which it cannot be used is the conversion between degrees Celsius and kelvins (or degrees Fahrenheit). Between degrees Celsius and kelvins, there is a constant difference rather than a constant ratio, while between degrees Celsius and degrees Fahrenheit there is neither a constant difference nor a constant ratio. There is, however, an affine transform ( x ↦ a x + b displaystyle xmapsto ax+b , rather than a linear transform x ↦ a x displaystyle xmapsto ax ) between them. For example, the freezing point of water is 0 °C and 32 °F, and a 5 °C change is the same as a 9 °F change. Thus, to convert from units of Fahrenheit to units of Celsius, one subtracts 32 °F (the offset from the point of reference), divides by 9 °F and multiplies by 5 °C (scales by the ratio of units), and adds 0 °C (the offset from the point of reference). Reversing this yields the formula for obtaining a quantity in units of Celsius from units of Fahrenheit; one could have started with the equivalence between 100 °C and 212 °F, though this would yield the same formula at the end. Hence, to convert the numerical quantity value of a temperature T[F] in degrees Fahrenheit to a numerical quantity value T[C] in degrees Celsius, this formula may be used: T[C] = (T[F] − 32) × 5/9. To convert T[C] in degrees Celsius to T[F] in degrees Fahrenheit, this formula may be used: T[F] = (T[C] × 9/5) + 32. Applications[edit]
x n , displaystyle x^ n , while the surface area, being ( n − 1 ) displaystyle (n-1) -dimensional, scales as x n − 1 . displaystyle x^ n-1 . Thus the volume of the n-ball in terms of the radius is C n r n , displaystyle C_ n r^ n , for some constant C n . displaystyle C_ n . Determining the constant takes more involved mathematics, but the form can be deduced and checked by dimensional analysis alone. Finance, economics, and accounting[edit] In finance, economics, and accounting, dimensional analysis is most commonly referred to in terms of the distinction between stocks and flows. More generally, dimensional analysis is used in interpreting various financial ratios, economics ratios, and accounting ratios. For example, the
Fluid mechanics[edit] Common dimensionless groups in fluid mechanics include:
R e = ρ V d / μ displaystyle mathrm Re =rho ,Vd/mu .
F r = V / g L . displaystyle mathrm Fr =V/ sqrt g,L . Euler number (Eu), used in problems in which pressure is of interest: E u = Δ p ρ V 2 . displaystyle mathrm Eu = frac Delta p rho V^ 2 .
M = V c , displaystyle mathrm M = frac V c , where : c is the local speed of sound. History[edit]
The origins of dimensional analysis have been disputed by
historians.[8][9] The 19th-century French mathematician Joseph Fourier
is generally credited with having made important contributions[10]
based on the idea that physical laws like F = ma should be independent
of the units employed to measure the physical variables. This led to
the conclusion that meaningful laws must be homogeneous equations in
their various units of measurement, a result which was eventually
formalized in the Buckingham π theorem. However, the first
application of dimensional analysis has been credited to the Italian
scholar
dim
Q = L a M b T c I d Θ e N f J g displaystyle text dim ~ Q = mathsf L ^ a mathsf M ^ b mathsf T ^ c mathsf I ^ d mathsf Theta ^ e mathsf N ^ f mathsf J ^ g where a, b, c, d, e, f, g are the dimensional exponents. Other physical quantities could be defined as the base quantities, as long as they form a linearly independent basis. For instance, one could replace the dimension of electrical current (I) of the SI basis with a dimension of electric charge (Q), since Q = IT. As examples, the dimension of the physical quantity speed v is dim v = length time = L T = L T − 1 displaystyle text dim ~v= frac text length text time = frac mathsf L mathsf T = mathsf LT ^ -1 and the dimension of the physical quantity force F is dim F = mass × acceleration = mass × length time 2 = M L T 2 = M L T − 2 displaystyle text dim ~F= text mass times text acceleration = text mass times frac text length text time ^ 2 = frac mathsf ML mathsf T ^ 2 = mathsf MLT ^ -2 The unit chosen to express a physical quantity and its dimension are related, but not identical concepts. The units of a physical quantity are defined by convention and related to some standard; e.g., length may have units of metres, feet, inches, miles or micrometres; but any length always has a dimension of L, no matter what units of length are chosen to express it. Two different units of the same physical quantity have conversion factors that relate them. For example, 1 in = 2.54 cm; in this case (2.54 cm/in) is the conversion factor, which is itself dimensionless. Therefore, multiplying by that conversion factor does not change the dimensions of a physical quantity. There are also physicists that have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity,[18] although this does not invalidate the usefulness of dimensional analysis. Mathematical properties[edit] Main article: Buckingham π theorem The dimensions that can be formed from a given collection of basic physical dimensions, such as M, L, and T, form an abelian group: The identity is written as 1; L0 = 1, and the inverse to L is 1/L or L−1. L raised to any rational power p is a member of the group, having an inverse of L−p or 1/Lp. The operation of the group is multiplication, having the usual rules for handling exponents (Ln × Lm = Ln+m). This group can be described as a vector space over the rational numbers, with for example dimensional symbol MiLjTk corresponding to the vector (i, j, k). When physical measured quantities (be they like-dimensioned or unlike-dimensioned) are multiplied or divided by one other, their dimensional units are likewise multiplied or divided; this corresponds to addition or subtraction in the vector space. When measurable quantities are raised to a rational power, the same is done to the dimensional symbols attached to those quantities; this corresponds to scalar multiplication in the vector space. A basis for such a vector space of dimensional symbols is called a set of base quantities, and all other vectors are called derived units. As in any vector space, one may choose different bases, which yields different systems of units (e.g., choosing whether the unit for charge is derived from the unit for current, or vice versa). The group identity 1, the dimension of dimensionless quantities, corresponds to the origin in this vector space. The set of units of the physical quantities involved in a problem correspond to a set of vectors (or a matrix). The nullity describes some number (e.g., m) of ways in which these vectors can be combined to produce a zero vector. These correspond to producing (from the measurements) a number of dimensionless quantities, π1, ..., πm . (In fact these ways completely span the null subspace of another different space, of powers of the measurements.) Every possible way of multiplying (and exponentiating) together the measured quantities to produce something with the same units as some derived quantity X can be expressed in the general form X = ∏ i = 1 m ( π i ) k i . displaystyle X=prod _ i=1 ^ m (pi _ i )^ k_ i ,. Consequently, every possible commensurate equation for the physics of the system can be rewritten in the form f ( π 1 , π 2 , . . . , π m ) = 0 . displaystyle f(pi _ 1 ,pi _ 2 ,...,pi _ m )=0,. Knowing this restriction can be a powerful tool for obtaining new insight into the system. Mechanics[edit] The dimension of physical quantities of interest in mechanics can be expressed in terms of base dimensions M, L, and T – these form a 3-dimensional vector space. This is not the only valid choice of base dimensions, but it is the one most commonly used. For example, one might choose force, length and mass as the base dimensions (as some have done), with associated dimensions F, L, M; this corresponds to a different basis, and one may convert between these representations by a change of basis. The choice of the base set of dimensions is thus a convention, with the benefit of increased utility and familiarity. The choice of base dimensions is not arbitrary, because the dimensions must form a basis: they must span the space, and be linearly independent. For example, F, L, M form a set of fundamental dimensions because they form a basis that is equivalent to M, L, T: the former can be expressed as [F = ML/T2], L, M, while the latter can be expressed as M, L, [T = (ML/F)1/2]. On the other hand, length, velocity and time (L, V, T) do not form a set of as base dimensions, for two reasons: There is no way to obtain mass – or anything derived from it, such as force – without introducing another base dimension (thus, they do not span the space). Velocity, being expressible in terms of length and time (V = L/T), is redundant (the set is not linearly independent). Other fields of physics and chemistry[edit] 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 number of moles of substance (the number of molecules divided by Avogadro's constant, ≈ 6.02 × 1023) is defined as a base unit, 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. The choice of the dimensions or even the number of dimensions to be used in different fields of physics is to some extent arbitrary, but consistency in use and ease of communications are common and necessary features. Polynomials and transcendental functions[edit] Scalar arguments to transcendental functions such as exponential, trigonometric and logarithmic functions, or to inhomogeneous polynomials, must be dimensionless quantities.[citation needed] (Note: this requirement is somewhat relaxed in Siano's orientational analysis described below, in which the square of certain dimensioned quantities are dimensionless.) While most mathematical identities about dimensionless numbers translate in a straightforward manner to dimensional quantities, care must be taken with logarithms of ratios: the identity log(a/b) = log a − log b, where the logarithm is taken in any base, holds for dimensionless numbers a and b, but it does not hold if a and b are dimensional, because in this case the left-hand side is well-defined but the right-hand side is not. Similarly, while one can evaluate monomials (xn) of dimensional quantities, one cannot evaluate polynomials of mixed degree with dimensionless coefficients on dimensional quantities: for x2, the expression (3 m)2 = 9 m2 makes sense (as an area), while for x2 + x, the expression (3 m)2 + 3 m = 9 m2 + 3 m does not make sense. However, polynomials of mixed degree can make sense if the coefficients are suitably chosen physical quantities that are not dimensionless. For example, 1 2 ⋅ ( − 32 foot second 2 ) ⋅ t 2 + ( 500 foot second ) ⋅ t . displaystyle frac 1 2 cdot left(-32 frac text foot text second ^ 2 right)cdot t^ 2 +left(500 frac text foot text second right)cdot t. This is the height to which an object rises in time t if the acceleration of gravity is 32 feet per second per second and the initial upward speed is 500 feet per second. It is not even necessary for t to be in seconds. For example, suppose t = 0.01 minutes. Then the first term would be 1 2 ⋅ ( − 32 foot second 2 ) ⋅ ( 0.01 minute ) 2 = 1 2 ⋅ − 32 ⋅ ( 0.01 2 ) ( minute second ) 2 ⋅ foot = 1 2 ⋅ − 32 ⋅ ( 0.01 2 ) ⋅ 60 2 ⋅ foot . displaystyle begin aligned & qquad frac 1 2 cdot left(-32 frac text foot text second ^ 2 right)cdot (0.01 text minute )^ 2 \[10pt]&= frac 1 2 cdot -32cdot (0.01^ 2 )left( frac text minute text second right)^ 2 cdot text foot \[10pt]&= frac 1 2 cdot -32cdot (0.01^ 2 )cdot 60^ 2 cdot text foot .end aligned Incorporating units[edit] 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.[19] Z = n × [ Z ] = n [ Z ] displaystyle Z=ntimes [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. A conversion factor, which is a ratio of like-dimensioned quantities and is equal to the dimensionless unity, is needed: 1 ft = 0.3048 m
displaystyle 1 mbox ft =0.3048 mbox m is identical to 1 = 0.3048 m 1 ft . displaystyle 1= frac 0.3048 mbox m 1 mbox ft . The factor 0.3048 m ft displaystyle 0.3048 frac mbox m mbox ft is identical to the dimensionless 1, so multiplying by this conversion factor changes nothing. Then when adding two quantities of like dimension, but expressed in different units, the appropriate conversion factor, which is essentially the dimensionless 1, is used to convert the quantities to identical units so that their numerical values can be added or subtracted. Only in this manner is it meaningful to speak of adding like-dimensioned quantities of differing units. Position vs displacement[edit] Main article: Affine space Some discussions of dimensional analysis implicitly describe all quantities as mathematical vectors. (In mathematics scalars are considered a special case of vectors[citation needed]; vectors can be added to or subtracted from other vectors, and, inter alia, multiplied or divided by scalars. If a vector is used to define a position, this assumes an implicit point of reference: an origin. While this is useful and often perfectly adequate, allowing many important errors to be caught, it can fail to model certain aspects of physics. A more rigorous approach requires distinguishing between position and displacement (or moment in time versus duration, or absolute temperature versus temperature change). Consider points on a line, each with a position with respect to a given origin, and distances among them. Positions and displacements all have units of length, but their meaning is not interchangeable: adding two displacements should yield a new displacement (walking ten paces then twenty paces gets you thirty paces forward), adding a displacement to a position should yield a new position (walking one block down the street from an intersection gets you to the next intersection), subtracting two positions should yield a displacement, but one may not add two positions. This illustrates the subtle distinction between affine quantities (ones modeled by an affine space, such as position) and vector quantities (ones modeled by a vector space, such as displacement). Vector quantities may be added to each other, yielding a new vector quantity, and a vector quantity may be added to a suitable affine quantity (a vector space acts on an affine space), yielding a new affine quantity. Affine quantities cannot be added, but may be subtracted, yielding relative quantities which are vectors, and these relative differences may then be added to each other or to an affine quantity. Properly then, positions have dimension of affine length, while displacements have dimension of vector length. To assign a number to an affine unit, one must not only choose a unit of measurement, but also a point of reference, while to assign a number to a vector unit only requires a unit of measurement. Thus some physical quantities are better modeled by vectorial quantities while others tend to require affine representation, and the distinction is reflected in their dimensional analysis. This distinction is particularly important in the case of temperature, for which the numeric value of absolute zero is not the origin 0 in some scales. For absolute zero, 0 K = −273.15 °C = −459.67 °F = 0 °R, but for temperature differences, 1 K = 1 °C ≠ 1 °F = 1 °R. (Here °R refers to the Rankine scale, not the Réaumur scale). Unit conversion for temperature differences is simply a matter of multiplying by, e.g., 1 °F / 1 K (although the ratio is not a constant value). But because some of these scales have origins that do not correspond to absolute zero, conversion from one temperature scale to another requires accounting for that. As a result, simple dimensional analysis can lead to errors if it is ambiguous whether 1 K means the absolute temperature equal to −273.15 °C, or the temperature difference equal to 1 °C. Orientation and frame of reference[edit] Similar to the issue of a point of reference is the issue of orientation: a displacement in 2 or 3 dimensions is not just a length, but is a length together with a direction. (This issue does not arise in 1 dimension, or rather is equivalent to the distinction between positive and negative.) Thus, to compare or combine two dimensional quantities in a multi-dimensional space, one also needs an orientation: they need to be compared to a frame of reference. This leads to the extensions discussed below, namely Huntley's directed dimensions and Siano's orientational analysis. Examples[edit] A simple example: period of a harmonic oscillator[edit] What is the period of oscillation T of a mass m attached to an ideal linear spring with spring constant k suspended in gravity of strength g? That period is the solution for T of some dimensionless equation in the variables T, m, k, and g. The four quantities have the following dimensions: T [T]; m [M]; k [M/T2]; and g [L/T2]. From these we can form only one dimensionless product of powers of our chosen variables, G 1 displaystyle G_ 1 = T 2 k / m displaystyle T^ 2 k/m [T2 · M/T2 / M = 1], and putting G 1 = C displaystyle G_ 1 =C for some dimensionless constant C 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 g does not occur in the group. It is easy to
see that it is impossible to form a dimensionless product of powers
that combines g with k, m, and T, because g is the only quantity that
involves the dimension L. This implies that in this problem the g is
irrelevant.
T = κ m k displaystyle T=kappa sqrt tfrac m k , for some dimensionless constant κ (equal to C displaystyle sqrt C from the original dimensionless equation). When faced with a case where dimensional analysis rejects a variable (g, here) that one intuitively expects to belong in a physical description of the situation, another possibility is that the rejected variable is in fact relevant, but that some other relevant variable has been omitted, which might combine with the rejected variable to form a dimensionless quantity. 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 κ. A more complex example: energy of a vibrating wire[edit] Consider the case of a vibrating wire of length ℓ (L) vibrating with an amplitude A (L). The wire has a linear density ρ (M/L) and is under tension s (ML/T2), and we want to know the energy E (ML2/T2) in the wire. Let π1 and π2 be two dimensionless products of powers of the variables chosen, given by π 1 = E / A s π 2 = ℓ / A . displaystyle begin aligned pi _ 1 &=E/As\pi _ 2 &=ell /A.end aligned The linear density of the wire is not involved. The two groups found can be combined into an equivalent form as an equation F ( E / A s , ℓ / A ) = 0 , displaystyle F(E/As,ell /A)=0, where F is some unknown function, or, equivalently as E = A s f ( ℓ / A ) , displaystyle E=Asf(ell /A), where f is some other unknown function. Here the unknown function
implies that our solution is now incomplete, but dimensional analysis
has given us something that may not have been obvious: the energy is
proportional to the first power of the tension. Barring further
analytical analysis, we might proceed to experiments to discover the
form for the unknown function f. But our experiments are simpler than
in the absence of dimensional analysis. We'd perform none to verify
that the energy is proportional to the tension. Or perhaps we might
guess that the energy is proportional to ℓ, and so infer that E =
ℓs. The power of dimensional analysis as an aid to experiment and
forming hypotheses becomes evident.
The power of dimensional analysis really becomes apparent when it is
applied to situations, unlike those given above, that are more
complicated, the set of variables involved are not apparent, and the
underlying equations hopelessly complex. Consider, for example, a
small pebble sitting on the bed of a river. If the river flows fast
enough, it will actually raise the pebble and cause it to flow along
with the water. At what critical velocity will this occur? Sorting out
the guessed variables is not so easy as before. 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.
A third example: demand versus capacity for a rotating disc[edit]
Consider the case of a thin, solid, parallel-sided rotating disc of
axial thickness t (L) and radius R (L). The disc has a density ρ
(M/L3), rotates at an angular velocity ω (T−1) and this leads to a
stress S (ML−1T−2) in the material. There is a theoretical linear
elastic solution, given by Lame, to this problem when the disc is thin
relative to its radius, the faces of the disc are free to move
axially, and the plane stress constitutive relations can be assumed to
be valid. As the disc becomes thicker relative to the radius then the
plane stress solution breaks down. If the disc is restrained axially
on its free faces then a state of plane strain will occur. However, if
this is not the case then the state of stress may only be determined
though consideration of three-dimensional elasticity and there is no
known theoretical solution for this case. An engineer might,
therefore, be interested in establishing a relationship between the
five variables.
demand/capacity = ρR2ω2/S thickness/radius or aspect ratio = t/R Dimensional Analysis and Numerical Experiments for a Rotating Disc Through the use of numerical experiments using, for example, the finite element method, the nature of the relationship between the two non-dimensional groups can be obtained as shown in the figure. As this problem only involves two non-dimensional groups, the complete picture is provided in a single plot and this can be used as a design/assessment chart for rotating discs[20] Extensions[edit] Huntley's extension: directed dimensions[edit] Huntley (Huntley 1967) has pointed out that it is sometimes productive to refine our concept of dimension. Two possible refinements are: The magnitude of the components of a vector are to be considered
dimensionally distinct. For example, rather than an undifferentiated
length dimension L, we may have Lx represent dimension in the
x-direction, and so forth. This requirement stems ultimately from the
requirement that each component of a physically meaningful equation
(scalar, vector, or tensor) must be dimensionally consistent.
As an example of the usefulness of the first refinement, suppose we wish to calculate the distance a cannonball travels when fired with a vertical velocity component V y displaystyle V_ mathrm y and a horizontal velocity component V x displaystyle V_ mathrm x , assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are then V x displaystyle V_ mathrm x , V y displaystyle V_ mathrm y , both dimensioned as LT−1, R, the distance travelled, having dimension L, and g the downward acceleration of gravity, with dimension LT−2. With these four quantities, we may conclude that the equation for the range R may be written: R ∝ V x a V y b g c . displaystyle Rpropto V_ text x ^ a ,V_ text y ^ b ,g^ c ., Or dimensionally L = ( L / T ) a + b ( L / T 2 ) c displaystyle mathsf L =( mathsf L / mathsf T )^ a+b ( mathsf L / mathsf T ^ 2 )^ c , from which we may deduce that a + b + c = 1 displaystyle a+b+c=1 and a + b + 2 c = 0 displaystyle a+b+2c=0 , which leaves one exponent undetermined. This is to be expected since we have two fundamental dimensions L and T, and four parameters, with one equation. If, however, we use directed length dimensions, then V x displaystyle V_ mathrm x will be dimensioned as LxT−1, V y displaystyle V_ mathrm y as LyT−1, R as Lx and g as LyT−2. The dimensional equation becomes: L x = ( L x / T ) a ( L y / T ) b ( L y / T 2 ) c displaystyle mathsf L _ mathrm x =( mathsf L _ mathrm x / mathsf T )^ a ,( mathsf L _ mathrm y / mathsf T )^ b ( mathsf L _ mathrm y / mathsf T ^ 2 )^ c , and we may solve completely as a = 1 displaystyle a=1 , b = 1 displaystyle b=1 and c = − 1 displaystyle c=-1 . The increase in deductive power gained by the use of directed length dimensions is apparent. In a similar manner, it is sometimes found useful (e.g., in fluid mechanics and thermodynamics) to distinguish between mass as a measure of inertia (inertial mass), and mass as a measure of quantity (substantial mass). For example, consider the derivation of Poiseuille's Law. We wish to find the rate of mass flow of a viscous fluid through a circular pipe. Without drawing distinctions between inertial and substantial mass we may choose as the relevant variables m ˙ displaystyle dot m the mass flow rate with dimension MT−1 p x displaystyle p_ text x the pressure gradient along the pipe with dimension ML−2T−2 ρ the density with dimension ML−3 η the dynamic fluid viscosity with dimension ML−1T−1 r the radius of the pipe with dimension L There are three fundamental variables so the above five equations will yield two dimensionless variables which we may take to be π 1 = m ˙ / η r displaystyle pi _ 1 = dot m /eta r and π 2 = p x ρ r 5 / m ˙ 2 displaystyle pi _ 2 =p_ mathrm x rho r^ 5 / dot m ^ 2 and we may express the dimensional equation as C = π 1 π 2 a = ( m ˙ η r ) ( p x ρ r 5 m ˙ 2 ) a displaystyle C=pi _ 1 pi _ 2 ^ a =left( frac dot m eta r right)left( frac p_ mathrm x rho r^ 5 dot m ^ 2 right)^ a where C and a are undetermined constants. If we draw a distinction between inertial mass with dimension M i displaystyle M_ text i and substantial mass with dimension M s displaystyle M_ text s , then mass flow rate and density will use substantial mass as the mass parameter, while the pressure gradient and coefficient of viscosity will use inertial mass. We now have four fundamental parameters, and one dimensionless constant, so that the dimensional equation may be written: C = p x ρ r 4 η m ˙ displaystyle C= frac p_ mathrm x rho r^ 4 eta dot m where now only C is an undetermined constant (found to be equal to π / 8 displaystyle pi /8 by methods outside of dimensional analysis). This equation may be solved for the mass flow rate to yield Poiseuille's law. Siano's extension: orientational analysis[edit] Huntley's extension has some serious drawbacks: It does not deal well with vector equations involving the cross product, nor does it handle well the use of angles as physical variables. It also is often quite difficult to assign the L, Lx, Ly, Lz, symbols to the physical variables involved in the problem of interest. He invokes a procedure that involves the "symmetry" of the physical problem. This is often very difficult to apply reliably: It is unclear as to what parts of the problem that the notion of "symmetry" is being invoked. Is it the symmetry of the physical body that forces are acting upon, or to the points, lines or areas at which forces are being applied? What if more than one body is involved with different symmetries? Consider the spherical bubble attached to a cylindrical tube, where one wants the flow rate of air as a function of the pressure difference in the two parts. What are the Huntley extended dimensions of the viscosity of the air contained in the connected parts? What are the extended dimensions of the pressure of the two parts? Are they the same or different? These difficulties are responsible for the limited application of Huntley's addition to real problems. Angles are, by convention, considered to be dimensionless variables, and so the use of angles as physical variables in dimensional analysis can give less meaningful results. As an example, consider the projectile problem mentioned above. Suppose that, instead of the x- and y-components of the initial velocity, we had chosen the magnitude of the velocity v and the angle θ at which the projectile was fired. The angle is, by convention, considered to be dimensionless, and the magnitude of a vector has no directional quality, so that no dimensionless variable can be composed of the four variables g, v, R, and θ. Conventional analysis will correctly give the powers of g and v, but will give no information concerning the dimensionless angle θ. Siano (1985-I, 1985-II) has suggested that the directed dimensions of Huntley be replaced by using orientational symbols 1x 1y 1z to denote vector directions, and an orientationless symbol 10. Thus, Huntley's Lx becomes L 1x with L specifying the dimension of length, and 1x specifying the orientation. Siano further shows that the orientational symbols have an algebra of their own. Along with the requirement that 1i−1 = 1i, the following multiplication table for the orientation symbols results: 1 0 1 x 1 y 1 z 1 0 1 0 1 x 1 y 1 z 1 x 1 x 1 0 1 z 1 y 1 y 1 y 1 z 1 0 1 x 1 z 1 z 1 y 1 x 1 0 displaystyle begin array ccccc &mathbf 1_ 0 &mathbf 1_ text x &mathbf 1_ text y &mathbf 1_ text z \hline mathbf 1_ 0 &1_ 0 &1_ text x &1_ text y &1_ text z \mathbf 1_ text x &1_ text x &1_ 0 &1_ text z &1_ text y \mathbf 1_ text y &1_ text y &1_ text z &1_ 0 &1_ text x \mathbf 1_ text z &1_ text z &1_ text y &1_ text x &1_ 0 end array Note that the orientational symbols form a group (the Klein four-group
or "Viergruppe"). In this system, scalars always have the same
orientation as the identity element, independent of the "symmetry of
the problem".
sin ( θ + π / 2 ) = cos ( θ ) displaystyle sin(theta +pi /2)=cos(theta ) is not dimensionally inconsistent since it is a special case of the sum of angles formula and should properly be written: sin ( a 1 z + b 1 z ) = sin ( a 1 z ) cos ( b 1 z ) + sin ( b 1 z ) cos ( a 1 z ) , displaystyle sin(a,1_ text z +b,1_ text z )=sin(a,1_ text z )cos(b,1_ text z )+sin(b,1_ text z )cos(a,1_ text z ), which for a = θ displaystyle a=theta and b = π / 2 displaystyle b=pi /2 yields sin ( θ 1 z + ( π / 2 ) 1 z ) = 1 z cos ( θ 1 z ) displaystyle sin(theta ,1_ text z +(pi /2),1_ text z )=1_ text z cos(theta ,1_ text z ) .
e i θ displaystyle e^ itheta ) which imply that the complex quantity i has an orientation equal to that of the angle it is associated with (1z in the above example). The assignment of orientational symbols to physical quantities and the requirement that physical equations be orientationally homogeneous can actually be used in a way that is similar to dimensional analysis to derive a little more information about acceptable solutions of physical problems. In this approach one sets up the dimensional equation and solves it as far as one can. If the lowest power of a physical variable is fractional, both sides of the solution is raised to a power such that all powers are integral. This puts it into "normal form". The orientational equation is then solved to give a more restrictive condition on the unknown powers of the orientational symbols, arriving at a solution that is more complete than the one that dimensional analysis alone gives. Often the added information is that one of the powers of a certain variable is even or odd. As an example, for the projectile problem, using orientational symbols, θ, being in the xy-plane will thus have dimension 1z and the range of the projectile R will be of the form: R = g a v b θ c which means L 1 x ∼ ( L 1 y T 2 ) a ( L T ) b 1 z c . displaystyle R=g^ a ,v^ b ,theta ^ c text which means mathsf L ,1_ mathrm x sim left( frac mathsf L ,1_ text y mathsf T ^ 2 right)^ a left( frac mathsf L mathsf T right)^ b ,1_ mathsf z ^ c ., Dimensional homogeneity will now correctly yield a = −1 and b = 2,
and orientational homogeneity requires that c be an odd integer. In
fact the required function of theta will be sin(θ)cos(θ) which is a
series of odd powers of θ.
It is seen that the Taylor series of sin(θ) and cos(θ) are
orientationally homogeneous using the above multiplication table,
while expressions like cos(θ) + sin(θ) and exp(θ) are not, and are
(correctly) deemed unphysical. It should be clear that the
multiplication rule used for the orientational symbols is not the same
as that for the cross product of two vectors. The cross product of two
identical vectors is zero, while the product of two identical
orientational symbols is the identity element.
Dimensionless concepts[edit]
Constants[edit]
Main article: Dimensionless quantity
The dimensionless constants that arise in the results obtained, such
as the C in the
κ displaystyle 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.
ξ displaystyle xi ) becomes larger and larger. Now, the correlation length is the relevant length scale related to critical phenomena, so one can, e.g., surmise on "dimensional grounds" that the non-analytical part of the free energy per lattice site should be ∼ 1 / ξ d displaystyle sim 1/xi ^ d where d displaystyle d is the dimension of the lattice.
It has been argued by some physicists, e.g., M. J. Duff,[18][21] that
the laws of physics are inherently dimensionless. The fact that we
have assigned incompatible dimensions to Length,
c → ∞ displaystyle crightarrow infty , ℏ → 0 displaystyle hbar rightarrow 0 and G → 0 displaystyle Grightarrow 0 . In problems involving a gravitational field the latter limit should be taken such that the field stays finite. Dimensional equivalences[edit] Following are tables of commonly occurring expressions in physics, related to the dimensions of energy, momentum, and force.[22][23][24] SI units[edit]
Expression Nomenclature Mechanical F d displaystyle Fd F = force, d = distance S / t ≡ P t displaystyle S/tequiv Pt S = action, t = time, P = power m v 2 ≡ p v ≡ p 2 / m displaystyle mv^ 2 equiv pvequiv p^ 2 /m m = mass, v = velocity, p = momentum I ω 2 ≡ L ω ≡ L 2 / I displaystyle Iomega ^ 2 equiv Lomega equiv L^ 2 /I L = angular momentum, I = moment of inertia, ω = angular velocity Thermal p V ≡ n R T ≡ k B T ≡ T S displaystyle pVequiv nRTequiv k_ B Tequiv TS p = pressure, T = temperature, S = entropy, kB = boltzmann constant, R = gas constant Waves I A t ≡ S A t displaystyle IAtequiv SAt I = wave intensity, S = Poynting vector Electromagnetic q ϕ displaystyle qphi q = electric charge, ϕ = electric potential (for changes this is voltage) ε E 2 V ≡ B 2 V / μ displaystyle varepsilon E^ 2 Vequiv B^ 2 V/mu E = electric field, B = magnetic field, ε= permittivity, μ = permeability, V = 3d volume p E ≡ m B ≡ I A displaystyle pEequiv mBequiv IA p = electric dipole moment, m = magnetic moment, A = area (bounded by a current loop), I = electric current in loop
Expression Nomenclature Mechanical m v ≡ F t displaystyle mvequiv Ft m = mass, v = velocity, F = force, t = time S / r ≡ L / r displaystyle S/requiv L/r S = action, L = angular momentum, r = displacement Thermal m ⟨ v 2 ⟩ displaystyle m sqrt langle v^ 2 rangle ⟨ v 2 ⟩ displaystyle sqrt langle v^ 2 rangle = root mean square velocity, m = mass (of a molecule) Waves ρ V v displaystyle rho Vv ρ = mass density, V = 3d volume, v phase velocity, Electromagnetic q A displaystyle qA A = magnetic vector potential
Expression Nomenclature Mechanical m a ≡ p / t displaystyle maequiv p/t m = mass, a = acceleration Thermal T δ S / δ r displaystyle Tdelta S/delta r S entropy, T = temperature, r = displacement (see entropic force) Waves ρ V v displaystyle rho Vv ρ = mass density, V = 3d volume, v phase velocity, Electromagnetic E q ≡ B q v displaystyle Eqequiv Bqv E = electric field, B = magnetic field, v = velocity, q = charge Natural units[edit] Main article: Natural units If c = ħ = 1, where c is the speed of light and ħ is the reduced Planck constant, and a suitable fixed unit of energy is chosen, then all quantities of length L, mass M and time T can be expressed (dimensionally) as a power of energy E, because length, mass and time can be expressed using speed v, action S, and energy E:[24] M = E / v 2 , L = S v / E , t = S / E displaystyle M=E/v^ 2 ,quad L=Sv/E,quad t=S/E though speed and action are dimensionless (v = c = 1 and S = ħ = 1) – so the only remaining quantity with dimension is energy. In terms of powers of dimensions: E n = M p L q T r = E p − q − r displaystyle mathsf E ^ n = mathsf M ^ p mathsf L ^ q mathsf T ^ r = mathsf E ^ p-q-r This is particularly useful in particle physics and high energy physics, in which case the energy unit is the electron volt (eV). Dimensional checks and estimates become very simple in this system. However, if electric charges and currents are involved, another unit to be fixed is for electric charge, normally the electron charge e though other choices are possible. Quantity p, q, r powers of energy n power of energy p q r n Action S 1 2 –1 0
See also[edit]
Related areas of math[edit] Covariance and contravariance of vectors Exterior algebra Geometric algebra Quantity calculus Notes[edit] ^ Goldberg, David (2006). Fundamentals of Chemistry (5th ed.).
McGraw-Hill. ISBN 0-07-322104-X.
^ Ogden, James (1999). The Handbook of Chemical Engineering. Research
& Education Association. ISBN 0-87891-982-1.
^ Dimensional Analysis or the Factor Label Method
^ Fourier, Joseph. Théorie analytique de la chaleur, Firmin Didot,
Paris, 1822.
^ JCGM 200:2012 International vocabulary of metrology – Basic and
general concepts and associated terms (VIM) Archived 2015-09-23 at the
Wayback Machine.
^ Cimbala and Cengel (2006), Fluid Mechanics: Fundamentals and
Applications, McGraw-Hill. Chapter 7: "Dimensional Analysis and
Modeling, Section 7-2: "Dimensional homogeneity" [1]
^ de Jong, Frits J.; Quade, Wilhelm (1967).
References[edit] Barenblatt, G. I. (1996), Scaling, Self-Similarity, and Intermediate
Asymptotics, Cambridge, UK: Cambridge University Press,
ISBN 0-521-43522-6
Bhaskar, R.; Nigam, Anil (1990), "Qualitative Physics Using
Dimensional Analysis", Artificial Intelligence, 45: 73–111,
doi:10.1016/0004-3702(90)90038-2
Bhaskar, R.; Nigam, Anil (1991), "Qualitative Explanations of Red
Giant Formation", The Astrophysical Journal, 372: 592–6,
Bibcode:1991ApJ...372..592B, doi:10.1086/170003
Boucher; Alves (1960), "Dimensionless Numbers", Chemical Engineering
Progress, 55: 55–64
Bridgman, P. W. (1922), Dimensional Analysis, Yale University Press,
ISBN 0-548-91029-4
Buckingham, Edgar (1914), "On Physically Similar Systems:
Illustrations of the Use of Dimensional Analysis", Physical Review, 4
(4): 345–376, Bibcode:1914PhRv....4..345B,
doi:10.1103/PhysRev.4.345
Drobot, S. (1953–1954), "On the foundations of dimensional analysis"
(PDF), Studia Mathematica, 14: 84–99
Gibbings, J.C. (2011), Dimensional Analysis, Springer,
ISBN 1-84996-316-9
Hart, George W. (1994), "The theory of dimensioned matrices", in
Lewis, John G., Proceedings of the Fifth SIAM Conference on Applied
Linear Algebra, SIAM, pp. 186–190,
ISBN 978-0-89871-336-7 As postscript
Hart, George W. (1995), Multidimensional Analysis: Algebras and
Systems for
External links[edit] The Wikibook Fluid
List of dimensions for variety of physical quantities Unicalc Live web calculator doing units conversion by dimensional analysis A C++ implementation of compile-time dimensional analysis in the Boost open-source libraries Buckingham’s pi-theorem Quantity System calculator for units conversion based on dimensional approach Units, quantities, and fundamental constants project dimensional analysis maps Bowley, Roger (2009). "[ ] Dimensional Analysis". Sixty Symbols. Brady Haran for the University of Nottingham. Converting units[edit] Unicalc Live web calculator doing units conversion by dimensional analysis Math Skills Review U.S. EPA tutorial A Discussion of Units Short Guide to Unit Conversions Canceling Units Lesson Chapter 11: Behavior of Gases Chemistry: Concepts and Applications, Denton Independent School District Air Dispersion Modeling Conversions and Formulas www.gnu.org/software/units free program, very practical v t e Systems of measurement Current General
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