BeginnerTopic 3 of 104Engineering Mathematics
Units and dimensional analysis
Checking that both sides of an equation have the same dimensions catches errors and helps derive relations.
In depth
Every physical quantity has dimensions such as length L, mass M and time T. Both sides of a valid equation must have identical dimensions, and only quantities with the same dimensions can be added. The Buckingham Pi theorem uses this to reduce variables to dimensionless groups.
Key points
- Convert to SI before calculating
- Arguments of sin, log and exp are dimensionless
- Dimensional consistency is necessary but not sufficient
ExampleSpeed has dimensions L/T, so v=at is valid only if a has L/T².
What is Units and dimensional analysis?
Checking that both sides of an equation have the same dimensions catches errors and helps derive relations.
Why is it important to understand units and dimensional analysis?
Dimensional analysis is a fast check that catches wrong formulas before you spend time on numbers, and it can even suggest the form of a relation.
If both sides of an equation have the right units, is it definitely correct?
No. Units only test consistency. A formula can match units and still be missing a numerical factor such as one half.
Why do many spacecraft and aircraft failures trace back to unit errors?
Mixing units, such as pound-force and newton or feet and metres, produces answers that are off by large factors, and the error passes unnoticed unless units are tracked.
How is Units and dimensional analysis used in practice?
Typical applications include checking a pipe-flow formula before programming it, sanity-checking a result from a finite-element model and deriving the period of a pendulum from length, mass and gravity.
What should I learn before Units and dimensional analysis?
Study "Significant figures" first (topic 2 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Units and dimensional analysis?
Convert to SI before calculating. Arguments of sin, log and exp are dimensionless. Dimensional consistency is necessary but not sufficient.
Can you give an example of Units and dimensional analysis?
Speed has dimensions L/T, so v=at is valid only if a has L/T².
What comes after Units and dimensional analysis?
The next topic is "Scientific notation", which builds directly on this one.
How difficult is Units and dimensional analysis?
It is classed as Beginner in Engineering Mathematics (topic 3 of 104). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.