Dimensions
- id: 1764770138
- Date: Dec. 3, 2025, 4:03 p.m.
- Author: Donald F. Elger
Goals
- Prioritize using dimensions.
- Skillfully use dimensions.
- Describe dimensions.
What
A dimension is a category whose members are units.
For example, the dimension length includes all possible units for quantifying length: meters, centimeters, kilometers, nanometers, inches, miles, feet, furlongs, and so on.
More Examples: This next table shows some common dimensions and units that are associated with each of these dimensions.
| Dimension | Examples of Units Associated with this dimension |
|---|---|
| Length | m, cm, mm, ft, mile, km |
| Speed | m/s, mph, km/h, inches/s |
| Mass | kg, lbm, ounce, ton |
| Time | s, min, hr, day |
| Temperature | K, °C, °F |
| Force | N, lbf, dyne |
| Energy | J, kWh, calorie, BTU |
Primary Dimensions and Secondary Dimensions
A secondary dimension is one that can be expressed in terms of 7 primary dimensions.
For example, speed (secondary dimension) can be expressed as length/time. Both length and time are primary dimensions.
The classical 7 primary dimensions (SI) are:
| Primary Dimension | Symbol | Representation | Unit (SI) |
|---|---|---|---|
| Length | L | size, distance | meter (m) |
| Mass | M | amount of matter | kilogram (kg) |
| Time | T | duration | second (s) |
| Electric Current | I | flow of charge | ampere (A) |
| Temperature | Θ | thermal state | kelvin (K) |
| Amount of Substance | N | number of particles | mole (mol) |
| Luminous Intensity | J | brightness | candela (cd) |
All other quantities are expressed in terms of these 7 by using equations. For example, Newton’s second law of motion F = ma relates force, mass, and acceleration. We say:
\[[F] = [ma] = [m][a]\]
This notation translates as: The dimensions of force are the product of the dimensions of mass times the dimensions of acceleration.
So force (a secondary dimension) can be expressed as mass*length/time^2.
Why?
Simplify the world.
- There are so many units that the set might as well be infinite.
- However, there are only about ~20 dimensions and only 7 primary dimensions.
- By organizing from primary dimensions → secondary dimensions → units, you put order on a huge, chaotic set.
Dimensions make remembering and using concepts and equations far easier because there are only a few dimensions to keep track of.
Understanding dimensions is required for dimensional analysis in fluid mechanics (and many other sciences). Dimensional analysis is essential for:
- experiment design
- scaling
- checking results
- understanding which variables matter
How
Through time and practice, learn to: 1. Break down concepts and equations into their units → secondary dimensions → primary dimensions. 2. Check equations for dimensional homogeneity. 3. Identify units, secondary dimensions, and primary dimensions of any quantity. 4. Think using dimensions (not units). - Example: Instead of thinking “this is 32 ft/s²,” think “this is acceleration → L/T².”
Primary Dimensions
A primary dimension (PD) is one of the fundamental building blocks used to describe all physical quantities. Every measurable quantity can be described using combinations of these PDs.
The classical 7 primary dimensions (SI) are:
| Primary Dimension | Symbol | Representation |
|---|---|---|
| Length | L | size, distance |
| Mass | M | amount of matter |
| Time | T | duration |
| Electric Current | I | flow of charge |
| Temperature | Θ | thermal state |
| Amount of Substance | N | number of particles |
| Luminous Intensity | J | brightness |
Almost all engineering and everyday quantities boil down to combinations of these seven.
Dimensional Homogeneity
- An equation is dimensionally homogeneous (DH) when:
- Every term in the equation has the same dimension.
- You can add quantities only if they belong to the same dimension.
- You can equate quantities only if they belong to the same dimension.
- Examples:
- v = x/t → (L/T) = L/T → DH
- s = ut + ½at² → every term has the dimension of length → consistent
- 5 meters + 3 seconds → nonsense (not DH)
- Dimensional homogeneity is the quickest way to:
- detect errors in equations
- check if a formula makes sense
- build new formulas
- scale experiments
Success Criteria for Dimensions
A learner has mastered this lesson when they can:
- Describe dimensions clearly (what, why, how).
- Relate primary dimensions ⇔ secondary dimensions ⇔ units.
- Use dimension notation to express any quantity.
- Find the primary dimensions of a parameter.
- Check an equation for dimensional homogeneity (DH).
- Describe DH and explain why it matters.
- Apply dimensional reasoning to check equations and results.
- Demonstrate high affect for dimensions by:
- reporting clarity, confidence, and usefulness
- expressing why dimensional thinking simplifies science
- showing willingness to use dimensions in problem solving
TwFs (Tasks with Feedback)
List three everyday quantities and classify each one by its dimension (not units).
Feedback
Good answers name real-world quantities and classify them by dimension.
Examples:
- Car speed → speed (L/T)
- Weight of a book → mass (M)
- Time to boil water → time (T)
If you give units instead of dimensions, convert units → dimensions.
Given the units below, state the dimension of each: - meters - newtons - °C - ft/s²
Feedback
Correct mapping:
- meters → length (L)
- newtons → force (M·L/T²)
- °C → temperature (Θ)
- ft/s² → acceleration (L/T²)
If you used units instead of dimensions, convert units → dimension symbols.
Explain in your own words what a primary dimension is.
Feedback
A strong answer makes these points:
- A primary dimension is a fundamental building block (L, M, T, I, Θ, N, J).
- All other dimensions can be expressed in terms of these.
- They are the “atoms” of dimensional reasoning.
Avoid giving examples of units (m, kg). Units are not primary dimensions.
Express the following secondary dimensions in terms of primary dimensions: 1. Speed 2. Force 3. Pressure
Feedback
Correct reductions:
- Speed → L/T
- Force → M·L/T²
- Pressure → M/(L·T²)
If you used units, convert units → dimensions → primary dimensions.
Write the dimension notation for the equation F = ma. Show each step.
Feedback
Correct breakdown:
- [F] = M·L/T²
- [m] = M
- [a] = L/T²
- So [F] = [m][a]
If symbols or units were misused, return to primary dimension definitions.
Check whether the equation v = at + x is dimensionally homogeneous.
Feedback
Reasoning:
- [v] = L/T
- [a]t = (L/T²)(T) = L/T
The equation adds terms with dimensions L/T and L.
Therefore, not DH.
Check whether the equation s = ut + ½at² is dimensionally homogeneous.
Feedback
Correct reasoning:
- [u]t = (L/T)(T) = L
- [a]t² = (L/T²)(T²) = L
All terms → L.
Therefore, the equation is DH.
A student claims: “Newton is a unit of mass.” Use dimensional reasoning to correct this misunderstanding.
Feedback
Clarifications:
- Newton is a unit of force, not mass.
- Force has dimension M·L/T².
- Mass has dimension M.
Mixing units and dimensions leads to conceptual errors.
Describe dimensional homogeneity: what it means, why it matters, and give an example of how it prevents mistakes.
Feedback
Key points:
- All terms in an equation must have the same dimensions.
- Ensures the equation is physically meaningful.
- Prevents mistakes like adding incompatible quantities.
- Helps catch algebraic or conceptual errors.
Convert the following variables to primary dimension notation: - kinetic energy: ½mv² - power: work/time - momentum: mv
Feedback
Correct reductions:
- Kinetic energy → M·L²/T²
- Power → M·L²/T³
- Momentum → M·L/T
If stuck, break each component into primary dimensions.
Explain why thinking in dimensions (L/T²) is more powerful than thinking in units (ft/s²).
Feedback
Good explanations include:
- Dimensions reveal structure and relationships.
- Units vary by system; dimensions remain constant.
- Dimensional thinking allows error-checking and concept comparison.
Rewrite this situation using dimensional thinking rather than units: “The pump produces 15 gallons per minute at 45 pounds per square inch.”
Feedback
Correct transformation:
- Flow rate → L³/T
- Pressure → M/(L·T²)
A good answer expresses the situation in dimension space, not unit space.
Create a real-world example where dimensional analysis helps check whether something “makes sense.”
Feedback
A strong example:
- Describes a concrete real situation.
- Shows how dimensional reasoning identifies what matters.
- Demonstrates the habit of checking reality using dimensions.
If the answer uses only units, convert them to dimensions.
Self-assess your affect: How confident, clear, and useful does dimensional thinking feel right now? Then explain one way you expect it to help you in future learning.
Feedback
A strong answer:
- Reports affect honestly.
- Identifies reasons for that affect.
- States at least one meaningful future use.
No wrong feelings; the goal is awareness. High clarity usually leads to rising affect.