In engineering and science, dimensional analysis of different physical quantities is the analysis of their physical dimension or quantity dimension, defined as a mathematical expression identifying the powers of the base quantities involved (such as length, mass, time, etc.), and tracking these dimensions as calculations or comparisons are performed. The concepts of dimensional analysis and quantity dimension were introduced by Joseph Fourier in 1822. Commensurable physical quantities have the same dimension and are of the same kind, so they can be directly compared to each other, even if they are expressed in differing units of measurement; e.g., metres and feet, grams and pounds, seconds and years. Incommensurable physical quantities have different dimensions, so can not be directly compared to each other, no matter what units they are expressed in, e.g. metres and grams, seconds and grams, metres and seconds. For example, asking whether a gram is larger than an hour is meaningless. Any physically meaningful equation or inequality must have the same dimensions on its left and right sides, a property known as dimensional homogeneity. Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on derived equations and computations. It also serves as a guide and constraint in deriving equations that may describe a physical system in the absence of a more rigorous derivation.
Formulation
The Buckingham π theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n − m 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 physical constants of nature. This may give insight into the fundamental properties of the system, as illustrated in the examples below. The dimension of a physical quantity can be expressed as a product of the base physical dimensions such as length, mass and time, each raised to an integer (and occasionally rational) power. The dimension of a physical quantity is more fundamental than some scale or unit used to express the amount of that physical quantity. For example, mass is a dimension, while the kilogram is a particular reference quantity chosen to express a quantity of mass. The choice of unit is arbitrary, and its choice is often based on historical precedent. Natural units, being based on only universal constants, may be thought of as being "less arbitrary". There are many possible choices of base physical dimensions. The SI standard selects the following dimensions and corresponding dimension symbols:
time (T), length (L), mass (M), electric current (I), absolute temperature (Θ), amount of substance (N) and luminous intensity (J). The symbols are by convention usually written in roman sans serif typeface. Mathematically, the dimension of the quantity Q is given by
dim Q = T a L b M c I d Θ e N f J g {\displaystyle \operatorname {dim} Q=T^{a}L^{b}M^{c}I^{d}{\mathsf {\Theta }}^{e}N^{f}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 basis – for instance, one could replace the dimension (I) of electric current of the SI basis with a dimension (Q) of electric charge, since Q = TI. A quantity that has only b ≠ 0 (with all other exponents zero) is known as a geometric quantity. A quantity that has only both a ≠ 0 and b ≠ 0 is known as a kinematic quantity. A quantity that has only all of a ≠ 0, b ≠ 0, and c ≠ 0 is known as a dynamic quantity. A quantity that has all exponents null is said to have dimension one. 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 and equal to 1. Therefore, multiplying by that conversion factor does not change either the dimensions nor the value of the physical quantity. There are also physicists who have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity, although this does not invalidate the usefulness of dimensional analysis.
Simple cases
As examples, the dimension of the physical quantity velocity v is
dim v = length time = L T = T − 1 L . {\displaystyle \operatorname {dim} v={\frac {\text{length}}{\text{time}}}={\frac {\mathsf {L}}{\mathsf {T}}}={\mathsf {T}}^{-1}{\mathsf {L}}.}
The dimension of the physical quantity acceleration a is
dim a = velocity time = T − 1 L T = T − 2 L . {\displaystyle \operatorname {dim} a={\frac {\text{velocity}}{\text{time}}}={\frac {{\mathsf {T}}^{-1}{\mathsf {L}}}{\mathsf {T}}}={\mathsf {T}}^{-2}{\mathsf {L}}.}
The dimension of the physical quantity force F is
dim F = mass × acceleration = M × T − 2 L = T − 2 L M . {\displaystyle \operatorname {dim} F={\text{mass}}\times {\text{acceleration}}={\mathsf {M}}\times {\mathsf {T}}^{-2}{\mathsf {L}}={\mathsf {T}}^{-2}{\mathsf {L}}{\mathsf {M}}.}
The dimension of the physical quantity pressure P is
dim P = force area = T − 2 L M L 2 = T − 2 L − 1 M . {\displaystyle \operatorname {dim} P={\frac {\text{force}}{\text{area}}}={\frac {{\mathsf {T}}^{-2}{\mathsf {L}}{\mathsf {M}}}{{\mathsf {L}}^{2}}}={\mathsf {T}}^{-2}{\mathsf {L}}^{-1}{\mathsf {M}}.}
The dimension of the physical quantity energy E is
dim E = force × displacement = T − 2 L M × L = T − 2 L 2 M . {\displaystyle \operatorname {dim} E={\text{force}}\times {\text{displacement}}={\mathsf {T}}^{-2}{\mathsf {L}}{\mathsf {M}}\times {\mathsf {L}}={\mathsf {T}}^{-2}{\mathsf {L}}^{2}{\mathsf {M}}.}
The dimension of the physical quantity power P is
dim P = energy time = T − 2 L 2 M T = T − 3 L 2 M . {\displaystyle \operatorname {dim} P={\frac {\text{energy}}{\text{time}}}={\frac {{\mathsf {T}}^{-2}{\mathsf {L}}^{2}{\mathsf {M}}}{\mathsf {T}}}={\mathsf {T}}^{-3}{\mathsf {L}}^{2}{\mathsf {M}}.}
The dimension of the physical quantity electric charge Q is
dim Q = current × time = T I . {\displaystyle \operatorname {dim} Q={\text{current}}\times {\text{time}}={\mathsf {T}}{\mathsf {I}}.}
The dimension of the physical quantity voltage V is
dim V = power current = T − 3 L 2 M I = T − 3 L 2 M I − 1 . {\displaystyle \operatorname {dim} V={\frac {\text{power}}{\text{current}}}={\frac {{\mathsf {T}}^{-3}{\mathsf {L}}^{2}{\mathsf {M}}}{\mathsf {I}}}={\mathsf {T^{-3}}}{\mathsf {L}}^{2}{\mathsf {M}}{\mathsf {I}}^{-1}.}
The dimension of the physical quantity capacitance C is
dim C = electric charge electric potential difference = T I T − 3 L 2 M I − 1 = T 4 L − 2 M − 1 I 2 . {\displaystyle \operatorname {dim} C={\frac {\text{electric charge}}{\text{electric potential difference}}}={\frac {{\mathsf {T}}{\mathsf {I}}}{{\mathsf {T}}^{-3}{\mathsf {L}}^{2}{\mathsf {M}}{\mathsf {I}}^{-1}}}={\mathsf {T^{4}}}{\mathsf {L^{-2}}}{\mathsf {M^{-1}}}{\mathsf {I^{2}}}.}
Rayleigh's method
In dimensional analysis, Rayleigh's method is a conceptual tool used in physics, chemistry, and engineering. It expresses a functional relationship of some variables in the form of an exponential equation. It was named after Lord Rayleigh. The method involves the following steps:
Gather all the independent variables that are likely to influence the dependent variable. If R is a variable that depends upon independent variables R1, R2, R3, ..., Rn, then the functional equation can be written as R = F(R1, R2, R3, ..., Rn). Write the above equation in the form R = C R1a R2b R3c ... Rnm, where C is a dimensionless constant and a, b, c, ..., m are arbitrary exponents. Express each of the quantities in the equation in some base units in which the solution is required. By using dimensional homogeneity, obtain a set of simultaneous equations involving the exponents a, b, c, ..., m. Solve these equations to obtain the values of the exponents a, b, c, ..., m. Substitute the values of exponents in the main equation, and form the non-dimensional parameters by grouping the variables with like exponents. As a drawback, Rayleigh's method does not provide any information regarding number of dimensionless groups to be obtained as a result of dimensional analysis.
Concrete numbers and base units
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 kilometres per hour or 1.4 kilometres per second. Compound relations with "per" are expressed with division, e.g. 60 km/h. Other relations can involve multiplication (often shown with a centered dot or juxtaposition), powers (like m2 for square metres), 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. 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 may be expressed as the product of mass (with unit kg) and acceleration (with unit m⋅s−2). The newton is defined as 1 N = 1 kg⋅m⋅s−2.
Percentages, derivatives and integrals
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 divides the dimension by the dimension of the variable that is differentiated with respect to. Thus:
position (x) has the dimension L (length); derivative of position with respect to time (dx/dt, velocity) has dimension T−1L—length from position, time due to the derivative; the second derivative (d2x/dt2 = d(dx/dt) / dt, acceleration) has dimension T−2L. Likewise, taking an integral adds the dimension of the variable one is integrating with respect to, but in the numerator.
force has the dimension T−2LM (mass multiplied by acceleration); the integral of force with respect to the distance (s) the object has travelled ( ∫ F d s {\displaystyle \textstyle \int F\ ds} , work) has dimension T−2L2M. In economics, one distinguishes between stocks and flows: a stock has a unit (say, widgets or dollars), while a flow is a derivative of a stock, and has a unit of the form of this unit divided by one of 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 the unit year, 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.
Dimensional homogeneity (commensurability)
The most basic rule of dimensional analysis is that of dimensional homogeneity.
However, the dimensions form an abelian group under multiplication, so:
For example, it makes no sense to ask whether 1 hour is more, the same, or less than 1 kilometre, as these have different dimensions, nor to add 1 hour to 1 kilometre. However, it makes sense to ask whether 1 mile is more, the same, or less than 1 kilometre, 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 T−2L2M, 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 unit. 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. For example, Newton's laws of motion must hold true whether distance is measured in miles or kilometres. This principle gives rise to the form that a conversion factor between two units that measure the same dimension must take multiplication by a simple constant. It also ensures equivalence; for example, if two buildings are the same height in feet, then they must be the same height in metres. For example, if one is calculating a speed, units must always combine to L/T; if one is calculating an energy, units must always combine to ML2/T2, etc. For example, the following formulae could be valid expressions for some energy:
E k = 1 2 m v 2 ; E = m c 2 ; E = p v ; E = h c / λ {\displaystyle E_{\text{k}}={\frac {1}{2}}mv^{2};~~E=mc^{2};~~E=pv;~~E=hc/\lambda }
if m is a mass, v and c are velocities, p is a momentum, h is the Planck constant, λ a length. On the other hand, if the units of the right hand side do not combine to [mass][length]2/[time]2, it cannot be a valid expression for some energy. Being homogeneous does not necessarily mean the equation will be true, since it does not take into account numerical factors. For example, E = mv2 could be or could not be the correct formula for the energy of a particle of mass m traveling at speed v, and one cannot know if hc/λ should be divided or multiplied by 2π.
Conversion factor
In dimensional analysis, a quotient 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 the unit. 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.
Applications Dimensional analysis is most often used in physics and chemistry – and in the mathematics thereof – but finds some applications outside of those fields as well.
Mathematics
A simple application of dimensional analysis to mathematics is in computing the form of the volume of an n-ball (the solid ball in n dimensions), or the area of its surface, the n-sphere: being an n-dimensional figure, the volume scales as xn, while the surface area, being (n − 1)-dimensional, scales as xn−1. Thus the volume of the n-ball in terms of the radius is Cnrn, for some constant Cn. Determining the constant takes more involved mathematics, but the form can be deduced and checked by dimensional analysis alone.
Finance, economics, and accounting
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 P/E ratio has dimensions of time (unit: year), and can be interpreted as "years of earnings to earn the price paid". In economics, debt-to-GDP ratio also has the unit year (debt has a unit of currency, GDP has a unit of currency/year). Velocity of money has a unit of 1/years (GDP/money supply has a unit of currency/year over currency): how often a unit of currency circulates per year. Annual continuously compounded interest rates and simple interest rates are often expressed as a percentage (adimensional quantity) while time is expressed as an adimensional quantity consisting of the number of years. However, if the time includes year as the unit of measure, the dimension of the rate is 1/year. Of course, there is nothing special (apart from the usual convention) about using year as a unit of time: any other time unit can be used. Furthermore, if rate and time include their units of measure, the use of different units for each is not problematic. In contrast, rate and time need to refer to a common period if they are adimensional. (Note that effective interest rates can only be defined as adimensional quantities.) In financial analysis, bond duration can be defined as (dV/dr)/V, where V is the value of a bond (or portfolio), r is the continuously compounded interest rate and dV/dr is a derivative. From the previous point, the dimension of r is 1/time. Therefore, the dimension of duration is time (usually expressed in years) because dr is in the "denominator" of the derivative.
Fluid mechanics
In fluid mechanics, dimensional analysis is performed to obtain dimensionless pi terms or groups. According to the principles of dimensional analysis, any prototype can be described by a series of these terms or groups that describe the behaviour of the system. Using suitable pi terms or groups, it is possible to develop a similar set of pi terms for a model that has the same dimensional relationships. In other words, pi terms provide a shortcut to developing a model representing a certain prototype. Common dimensionless groups in fluid mechanics include:
Reynolds number (Re), generally important in all types of fluid problems: R e = ρ u d μ . {\displaystyle \mathrm {Re} ={\frac {\rho \,ud}{\mu }}.}
Froude number (Fr), modeling flow with a free surface: F r = u g L . {\displaystyle \mathrm {Fr} ={\frac {u}{\sqrt {g\,L}}}.}
Euler number (Eu), used in problems in which pressure is of interest: E u = Δ p ρ u 2 . {\displaystyle \mathrm {Eu} ={\frac {\Delta p}{\rho u^{2}}}.}
Mach number (Ma), important in high speed flows where the velocity approaches or exceeds the local speed of sound: M a = u c , {\displaystyle \mathrm {Ma} ={\frac {u}{c}},} where c is the local speed of sound.
History The origins of dimensional analysis have been disputed by historians. The first written application of dimensional analysis has been credited to François Daviet, a student of Joseph-Louis Lagrange, in a 1799 article at the Turin Academy of Science. This led to the conclusion that meaningful laws must be homogeneous equations in their various units of measurement, a result which was eventually later formalized in the Buckingham π theorem. Simeon Poisson also treated the same problem of the parallelogram law by Daviet, in his treatise of 1811 and 1833 (vol I, p. 39). In the second edition of 1833, Poisson explicitly introduces the term dimension instead of the Daviet homogeneity. In 1822, the important Napoleonic scientist Joseph Fourier made the first credited important contributions based on the idea that physical laws like F = ma should be independent of the units employed to measure the physical variables. James Clerk Maxwell and Fleeming Jenkin played a major role in establishing modern use of dimensional analysis by distinguishing mass, length, and time as fundamental units, while referring to other units as derived. Although Maxwell defined length, time and mass to be "the three fundamental units", he also noted that gravitational mass can be derived from length and time by assuming a form of Newton's law of universal gravitation in which the gravitational constant G is taken as unity, thereby defining M = T−2L3. By assuming a form of Coulomb's law in which the Coulomb constant ke is taken as unity, Maxwell then determined that the dimensions of an electrostatic unit of charge were Q = T−1L3/2M1/2, which, after substituting his M = T−2L3 equation for mass, results in charge having the same dimensions as mass, viz. Q = T−2L3. Dimensional analysis is also used to derive relationships between the physical quantities that are involved in a particular phenomenon that one wishes to understand and characterize. It was used for the first time in this way in 1872 by Lord Rayleigh, who was trying to understand why the sky is blue. Rayleigh first published the technique in his 1877 book The Theory of Sound. The original meaning of the word dimension, in Fourier's Theorie de la Chaleur, was the numerical value of the exponents of the base units. For example, acceleration was considered to have the dimension 1 with respect to the unit of length, and the dimension −2 with respect to the unit of time. This was slightly changed by Maxwell, who said the dimensions of acceleration are T−2L, instead of just the exponents.
Examples
A simple example: period of a harmonic oscillator
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, G1 = T2k/m (T2 · M/T2 / M = 1), and putting G1 = 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. 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. Dimensional analysis can sometimes yield strong statements about the irrelevance of some quantities in a problem, or the need for additional parameters. If we have chosen enough variables to properly describe the problem, then from this argument we can conclude that the period of the mass on the spring is independent of g: it is the same on the earth or the moon. The equation demonstrating the existence of a product of powers for our problem can be written in an entirely equivalent way: 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
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 (LM/T2), and we want to know the energy E (L2M/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}&={\frac {E}{As}}\\\pi _{2}&={\frac {\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\left({\frac {E}{As}},{\frac {\ell }{A}}\right)=0,}
where F is some unknown function, or, equivalently as
E = A s f ( ℓ A ) , {\displaystyle E=Asf\left({\frac {\ell }{A}}\right),}
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
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 (T−2L−1M) 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. Dimensional analysis for this case leads to the following (5 − 3 = 2) non-dimensional groups:
demand/capacity = ρR2ω2/S thickness/radius or aspect ratio = t/R 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,