Dimensional analysis: Difference between revisions

In [[engineering]] and [[science]], '''dimensional analysis''' is the analysis of the relationships between different [[Physical quantity|physical quantities]] by identifying their [[base quantity|base quantities]] (such as [[length]], [[mass]], [[time]], and [[electric current]]) and [[units of measurement]] (such as metersmetres and grams) and tracking these dimensions as calculations or comparisons are performed. The term dimensional analysis is also used to refer to [[conversion of units]] from one dimensional unit to another, which can be used to evaluate scientific formulae.

'''''Commensurable''''' physical quantities are of the same [[Kind of quantity|kind]] and have the same dimension, and can be directly compared to each other, even if they are expressed in differing units of measurement; e.g., metersmetres and feet, gallonsgrams and literspounds, seconds and years. ''Incommensurable'' physical [[Quantity|quantities]] are of different [[Kind of quantity|kinds]] and have different dimensions, and 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 (mathematics)|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 [[Formal proof|derived]] equations and [[Computation|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.

The dimension of the physical quantity [[electric potential differencevoltage]] {{math|''V''}} is

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.<ref>{{Cite book |last1=de Jong |first1=Frits J. |url=https://archive.org/details/dimensionalanaly0000jong |title=Dimensional analysis for economists |last2=Quade |first2=Wilhelm |publisher=North Holland |year=1967 |page=[https://archive.org/details/dimensionalanaly0000jong/page/28 28] |url-access=registration}}</ref> 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 atwo unitunits that measuresmeasure 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.

A simple application of dimensional analysis to mathematics is in computing the form of the [[N-sphere#Volume of the n-ball|volume of an {{math|''n''}}-ball]] (the solid ball in ''n'' dimensions), or the area of its surface, the [[n-sphere|{{math|''n''}}-sphere]]: being an {{math|''n''}}-dimensional figure, the volume scales as {{math|''x''{{sup|''n''}}}}, while the surface area, being {{math|(''n'' − 1)}}-dimensional, scales as {{math|''x''{{sup|''n''−1}}}}. Thus the volume of the {{math|''n''}}-ball in terms of the radius is {{math|''C''{{isupsub|''n''}}''r''{{isup|''n''}}}}, for some constant {{math|''C''{{isupsub|''n''}}}}. Determining the constant takes more involved mathematics, but the form can be deduced and checked by dimensional analysis alone.

[[James Clerk Maxwell]] 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.<ref name="maxwell">{{Citation |last=Roche |first=John J |title=The Mathematics of Measurement: A Critical History |page=203 |year=1998 |publisher=Springer |isbn=978-0-387-91581-4 |url= https://books.google.com/books?id=eiQOqS-Q6EkC&pg=PA203|quote = Beginning apparently with Maxwell, mass, length and time began to be interpreted as having a privileged fundamental character and all other quantities as derivative, not merely with respect to measurement, but with respect to their physical status as well.}}</ref> 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]] {{math|''G''}} is taken as [[1|unity]], thereby defining {{nowrap|1=M = T⁻²L³}}.<ref name="maxwell2">{{Citation |last=Maxwell |first=James Clerk |title=A Treatise on Electricity and Magnetism |page=4 |year=1873}}</ref> By assuming a form of [[Coulomb's law]] in which the [[Coulomb constant]] ''k''_e is taken as unity, Maxwell then determined that the dimensions of an electrostatic unit of charge were {{nowrap|1=Q = T⁻¹L^3/2M^1/2}},<ref name="maxwell3">{{Citation |last= Maxwell |first=James Clerk |title=A Treatise on Electricity and Magnetism |series=Clarendon Press series |page=45 |year=1873 |publisher=Oxford |hdl=2027/uc1.l0065867749 |hdl-access=free}}</ref> which, after substituting his {{nowrap|1=M = T⁻²L³}} equation for mass, results in charge having the same dimensions as mass, viz. {{nowrap|1=Q = T⁻²L³}}.

Some discussions of dimensional analysis implicitly describe all quantities as mathematical vectors. (In mathematics scalars are considered a special case of vectors;{{citation needed|date=September 2013}} 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 (mathematics)|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).

[[Angle]]s are, by convention, considered to be dimensionless quantities (although the wisdom of this is contested <ref>{{ cite journal | title=Angles in the SI: a detailed proposal for solving the problem | year=2021 | pages=053002 | journal=Metrologia | doi=10.1088/1681-7575/ac023f | volume=58 | issue=5 | url=http://dx.doi.org/10.1088/1681-7575/ac023f | last1=Quincey | first1= Paul | arxiv=2108.05704 }}</ref>) . As an example, consider again the projectile problem in which a point mass is launched from the origin {{math|1=(''x'', ''y'') = (0, 0)}} at a speed {{math|''v''}} and angle {{math|''θ''}} above the ''x''-axis, with the force of gravity directed along the negative ''y''-axis. It is desired to find the range {{math|''R''}}, at which point the mass returns to the ''x''-axis. Conventional analysis will yield the dimensionless variable {{math|1=''π'' = ''R'' ''g''/''v''²}}, but offers no insight into the relationship between {{math|''R''}} and {{math|''θ''}}.