Dimensional Analysis

Pronunciation: /dɪˈmɛn ʃə nl əˈnæl ə sɪs/ ?

Mathematical equations used in science contain dimensions such as meters or seconds. Dimensional analysis is a tool for verifying these equations.[1] The dimensions, or units of measure, on both sides of an equation must agree for an equation to be valid.

A dimension is not the same as a unit of measure. A unit of measure may have multiple dimensions such as m/s (velocity), or m/s2 (acceleration).

Units of Measure and Their Dimensions
Unit of MeasureSymbolDimension
distancemmeters
kmkilometers
ftfeet
mimiles
timesseconds
hhours
msmilliseconds
masskgkilogram
velocitym/smeters per second
mi/hmiles per hour
km/hkilometers per hour
accelerationm/s2meters per second squared
mi/h2miles per hour squared
km/h2kilometers per hour squared
energy (joules)kg·m2/s2kilogram meters squared per second squared
force (newtons)kg·m/s2kilogram meters per second squared
Table 1. Underlined dimensions are the SI (Système Internationale) units. Use of SI System units is preferred over other, non-SI units.

Examples

  1. a = 3d/t2 where a is acceleration, d is distance, and t is time. Is this equation valid? To validate this equation, change the variables to units of measure. Use the table of units of measure as a guide. Constants that are not exponents can be ignored.
    • The dimensions of acceleration are m/s2.
    • The dimension of distance is m.
    • The dimension of time is s.
    So a = 3d/t2 becomes m/s2 = m/s2. Dimensional analysis does not show any problem with this equation.
  2. v = k·d/(3t) where v is velocity, k is mass, d is distance, and t is time. Check the validity of this equation using dimensional analysis.
    • The dimensions of velocity are m/s.
    • The dimension of mass is kg.
    • The dimension of distance is m.
    • The dimension of time is s.
    So v = k·d/(3t) becomes m/s = kg·m/s. The dimensions on both sides of the equation are not the same. This equation is invalid.

check mark Understanding Check

The variables in this understanding check have the following meanings:

  • d: distance
  • t: time
  • v: velocity
  • a: acceleration
  • m: mass
  • g: energy
  • f: force

Check the validity of each equation using dimensional analysis. Then click the 'Click for answer' button to see the correct answer.

blank space1. d = t·a Click for Answer
blank space2. g = f·d Click for Answer
blank space3. v = f·t/d Click for Answer
blank space4. m·f = g Click for Answer

References

  1. dimensional analysis. merriam-webster.com. Encyclopedia Britannica. (Accessed: 2010-01-23). http://www.merriam-webster.com/dictionary/dimensional analysis.
  2. Bridgman, P. W.. Dimensional Analysis. Yale University Press, 1922. http://www.archive.org/stream/dimensionalanal00bridgoog.
  3. Gloria P Craig. Quick Guide to Solving Problems Using Dimensional Analysis, 1st edition. Lippincott Williams & Wilkins, January 1, 2003.

More Information

  • Dimensional Analysis. Department of Physics, University of Guelph. 2009-03-12. http://www.physics.uoguelph.ca/tutorials/dimanaly/.

Printed Resources

Cite this article as:


Dimensional Analysis. 2010-01-23. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/d/dimensionalanalysis.html.

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2010-01-23: Added "References" (McAdams, David.)
2008-06-07: Corrected spelling (McAdams, David.)
2008-05-02: Initial version (McAdams, David.)

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