Editing Dimensional analysis (section)

=== A simple example: period of a harmonic oscillator ===
What is the period of [[Harmonic oscillator|oscillation]] {{math|''T''}} of a mass {{mvar|m}} attached to an ideal linear spring with spring constant {{math|''k''}} suspended in gravity of strength {{math|''g''}}? That period is the solution for {{math|''T''}} of some dimensionless equation in the variables {{math|''T''}}, {{math|''m''}}, {{math|''k''}}, and {{math|''g''}}.
The four quantities have the following dimensions:  {{mvar|T}}  [T];  {{mvar|m}}  [M]; {{mvar|k}} [M/T<sup>2</sup>]; and  {{math|''g''}} [L/T<sup>2</sup>]. From these we can form only one dimensionless product of powers of our chosen variables, {{math|1=''G''{{sub|1}} = ''T''{{isup|2}}''k''/''m''}} {{nowrap|1=[T<sup>2</sup> · M/T<sup>2</sup> / M = 1]}}, and putting {{math|1=''G''{{sub|1}} = ''C''}} for some dimensionless constant {{math|''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 (mathematics)|group]].  They are often called [[dimensionless number]]s as well.

The variable {{mvar|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 {{mvar|g}} with {{mvar|k}}, {{mvar|m}}, and {{mvar|T}}, because {{mvar|g}} is the only quantity that involves the dimension L. This implies that in this problem the {{math|''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 {{math|''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: {{tmath|1=T = \kappa \sqrt\tfrac{m}{k} }}, for some dimensionless constant {{math|''κ''}} (equal to <math>\sqrt{C}</math> from the original dimensionless equation).

When faced with a case where dimensional analysis rejects a variable ({{math|''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 {{math|''κ''}}.