Dimensional analysis


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, 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 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 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 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:
The symbols are by convention usually written in roman sans serif typeface. Mathematically, the dimension of the quantity is given by
where,,,,,, 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 of electric current of the SI basis with a dimension of electric charge, since.
A quantity that has only is known as a geometric quantity. A quantity that has only both and is known as a kinematic quantity. A quantity that has only all of,, and 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, ; 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 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 is
The dimension of the physical quantity acceleration is
The dimension of the physical quantity force is
The dimension of the physical quantity pressure is
The dimension of the physical quantity energy is
The dimension of the physical quantity power is
The dimension of the physical quantity electric charge is
The dimension of the physical quantity voltage is
The dimension of the physical quantity capacitance is

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:
  1. Gather all the independent variables that are likely to influence the dependent variable.
  2. If is a variable that depends upon independent variables,,,...,, then the functional equation can be written as.
  3. Write the above equation in the form, where is a dimensionless constant and,,,..., are arbitrary exponents.
  4. Express each of the quantities in the equation in some base units in which the solution is required.
  5. By using dimensional homogeneity, obtain a set of simultaneous equations involving the exponents,,,...,.
  6. Solve these equations to obtain the values of the exponents,,,...,.
  7. 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, powers, 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, 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 is a unit of force, which may be expressed as the product of mass and acceleration. The newton is defined as.

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.
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 has the dimension L ;
  • derivative of position with respect to time has dimension T−1L—length from position, time due to the derivative;
  • the second derivative has dimension.
Likewise, taking an integral adds the dimension of the variable one is integrating with respect to, but in the numerator.
  • force has the dimension ;
  • the integral of force with respect to the distance the object has travelled has dimension.
In economics, one distinguishes between stocks and flows: a stock has a unit, while a flow is a derivative of a stock, and has a unit of the form of this unit divided by one of time.
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 divided by annual GDP —but one may argue that, in comparing a stock to a flow, annual GDP should have dimensions of currency/time 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, and denote, respectively, the mass of some man, the mass of a rat and the length of that man, the dimensionally homogeneous expression is meaningful, but the heterogeneous expression is meaningless. However, 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, 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 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:
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 2/2, 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, 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''.