# Dimensional Analysis

Pronunciation: /dɪˈmɛn.ʃə.nl əˈnæl.ə.sɪs/ Explain

Mathematical equations used in science contain dimensions such as meters or seconds. Dimensional analysis is a tool for verifying these equations. 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 .

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 in red are the SI (Système Internationale) units. Use of SI System units are 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.

### 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. 1. d = t·a Click for AnswerInvalid. The dimensions are m = s·m/s2 which simplifies to m = m/s. 2. g = f·d Click for AnswerValid. The dimensions are kg·m2 / s2 = (kg·m / s2)·m which simplifies to kg·m2 / s2 = kg·m2 / s2. 3. v = f·t / d Click for AnswerValid. The dimensions are m / s = (kg·m / s2·s/m which simplifies to m/s = m/s. 4. m·f = g Click for AnswerInvalid. The dimensions are kg·(kg·m / s2) = kg·m2 / s2 which simplifies to kg2·m / s2 = kg·m2/s2.
1. McAdams, David E.. All Math Words Dictionary, dimensional analysis. 2nd Classroom edition 20150108-4799968. pg 61. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. dimensional analysis. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 7/3/2018. http://www.merriam-webster.com/dictionary/dimensional analysis. Buy the book
3. Bridgman, P. W.. Dimensional Analysis. www.archive.org. Yale University Press. 1922. Last Accessed 7/3/2018. http://www.archive.org/stream/dimensionalanal00bridgoog. Buy the book
4. Gloria P Craig. Quick Guide to Solving Problems Using Dimensional Analysis. 1st edition. Lippincott Williams & Wilkins. January 1, 2003. Last Accessed 7/3/2018. Buy the book

• Dimensional Analysis. Department of Physics, University of Guelph. 3/12/2009. http://www.physics.uoguelph.ca/tutorials/dimanaly/.

McAdams, David E. Dimensional Analysis. 4/19/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/d/dimensionalanalysis.html.

### Revision History

4/19/2019: Updated equations and expressions to the new format (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)