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/s^{2}
(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/s^{2} | meters per second squared |
mi/h^{2} | miles per hour squared |
km/h^{2} | kilometers per hour squared |
energy (joules) | kg·m^{2}/s^{2} | kilogram meters squared per second squared |
force (newtons) | kg·m/s^{2} | 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
- a = 3d / t^{2}
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/s^{2}.
- The dimension of distance is m.
- The dimension of time is s.
So a = 3d / t^{2}
becomes m/s^{2} = m/s^{2}. Dimensional analysis does not
show any problem with this equation.
- 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/s^{2} which simplifies to
m = m/s. |
2. |
g = f·d |
Click for AnswerValid. The dimensions are
kg·m^{2} / s^{2} =
(kg·m / s^{2})·m which simplifies to
kg·m^{2} / s^{2} = kg·m^{2}
/ s^{2}. |
3. |
v = f·t / d |
Click for AnswerValid. The dimensions are m / s = (kg·m / s^{2}·s/m which simplifies to m/s = m/s. |
4. |
m·f = g |
Click for AnswerInvalid. The dimensions are
kg·(kg·m / s^{2}) = kg·m^{2}
/ s^{2} which simplifies to kg^{2}·m / s^{2}
= kg·m^{2}/s^{2}. |
References
- 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
- 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
- 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
- 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
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.)