Dimensional Analysis

en · official · sha256:b56e633ebfab3ea113ad65221720c35fc003348fa85680925c72ea51394513cd

<list id="fs-id1168328228670" list-type="bulleted" bullet-style="bullet"> <item>Every term in an expression must have the same dimensions; it does not make sense to add or subtract quantities of differing dimension (think of the old saying: “You can’t add apples and oranges”). In particular, the expressions on each side of the equality in an equation must have the same dimensions.</item> <item>The arguments of any of the standard mathematical functions such as trigonometric functions (such as sine and cosine), logarithms, or exponential functions that appear in the equation must be dimensionless. These functions require pure numbers as inputs and give pure numbers as outputs.</item> </list>

Referencing norms: ClaimedProductMatches, EquationConsistentWhenNoTermDiffers, EquationInconsistentWhenTermDiffers, TermDiffersFromReference, TranscendentalArgumentMustBeDimensionless, argument_of_transcendental, claimed_product, equation_presented, reference_term, term_of

en · official · sha256:8bcad42fd10cb106d8e153ff96414b9b01f1efc0fa4ff84ac1541f1301c93333

<definition id="fs-id1168327872808"> <term>dimension</term> <meaning id="fs-id1168327908525">expression of the dependence of a physical quantity on the base quantities as a product of powers of symbols representing the base quantities; in general, the dimension of a quantity has the form <m:math><m:mrow><m:msup><m:mtext>L</m:mtext><m:mtext>a</m:mtext></m:msup><m:msup><m:mtext>M</m:mtext><m:mtext>b</m:mtext></m:msup><m:msup><m:mtext>T</m:mtext><m:mtext>c</m:mtext></m:msup><m:msup><m:mtext>I</m:mtext><m:mtext>d</m:mtext></m:msup><m:msup><m:mtext>Θ</m:mtext><m:mtext>e</m:mtext></m:msup><m:msup><m:mtext>N</m:mtext><m:mtext>f</m:mtext></m:msup><m:msup><m:mtext>J</m:mtext><m:mtext>g</m:mtext></m:msup></m:mrow></m:math> for some powers a, b, c, d, e, f, and g.</meaning> </definition>

Referencing norms: dimension_exponent

en · official · sha256:345c4265c2a3bbe020cc9c4a20f41a50c4d6453a6d49aa25b5c64a0d3b3844b8

<definition id="fs-id1168328321073"> <term>dimensionless</term> <meaning id="fs-id1168327949717">quantity with a dimension of <m:math><m:mrow><m:msup><m:mtext>L</m:mtext><m:mn>0</m:mn></m:msup><m:msup><m:mtext>M</m:mtext><m:mn>0</m:mn></m:msup><m:msup><m:mtext>T</m:mtext><m:mn>0</m:mn></m:msup><m:msup><m:mtext>I</m:mtext><m:mn>0</m:mn></m:msup><m:msup><m:mtext>Θ</m:mtext><m:mn>0</m:mn></m:msup><m:msup><m:mtext>N</m:mtext><m:mn>0</m:mn></m:msup><m:msup><m:mtext>J</m:mtext><m:mn>0</m:mn></m:msup><m:mo>=</m:mo><m:mn>1</m:mn><m:mo>;</m:mo></m:mrow></m:math> also called quantity of dimension 1 or a pure number</meaning> </definition>

Referencing norms: TermWithoutNonzeroExponentIsDimensionless

en · official · sha256:0a727ab95588c37848807e3e93d26e85c4a4bddff0a83da52b156fa915fd6162

<para id="fs-id1168327924785">If either of these rules is violated, an equation is not dimensionally consistent and cannot possibly be a correct statement of physical law. This simple fact can be used to check for typos or algebra mistakes, to help remember the various laws of physics, and even to suggest the form that new laws of physics might take. This last use of dimensions is beyond the scope of this text, but is something you may learn later in your academic career.</para>

Referencing norms: ConsistentMayBeLaw, InconsistentCannotBeLaw