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.^[3] 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 Measure	Symbol	Dimension
distance	m	meters
	km	kilometers
	ft	feet
	mi	miles
time	s	seconds
	h	hours
	ms	milliseconds
mass	kg	kilogram
velocity	m/s	meters per second
	mi/h	miles per hour
	km/h	kilometers per hour
acceleration	m/s2	meters per second squared
	mi/h2	miles per hour squared
	km/h2	kilometers per hour squared
energy (joules)	kg·m2/s2	kilogram meters squared per second squared
force (newtons)	kg·m/s2	kilogram 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.

References

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

More Information

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

Cite this article as:

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.

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

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.)
7/4/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
1/23/2010: Added "References". (McAdams, David E.)
6/7/2008: Corrected spelling. (McAdams, David E.)
5/2/2008: Initial version. (McAdams, David E.)