What this tool covers
Use a single screen to confirm unit algebra, show substitutions for students, and derive Buckingham Π groups for experiments.
- Expand any unit expression to the SI base vector and compute the scale factor k.
- Analyse equations term by term, ensuring additions are homogeneous and function arguments are dimensionless.
- Build Π groups by solving the null space of the dimension matrix with integer exponents.
- Share results through CSV export or a copyable URL that stores the current state.
Interactive calculator
Choose a mode, enter your variables, then review the annotated steps before exporting results.
Results
How it’s calculated
How to run a dimensional-analysis check
Start by listing every variable with a unit you trust, then use the unit and formula modes together. The unit mode confirms scale factors, while the formula mode tells you whether each side of the equation is dimensionally consistent.
What to check first
Normalize symbols and units before you compare terms. A mismatch between metres and millimetres, or between force and pressure, is easier to catch before you build the full expression.
How to interpret the output
Read the base-dimension vector and the step list together. Matching vectors mean the equation is dimensionally valid, but they do not prove the numerical constant or physical model is correct.
Common mistakes to avoid
- Checking only one side of the equation instead of every summed term.
- Feeding trig, exponential, or logarithmic functions a quantity that still has units.
- Mixing SI prefixes without verifying the scale factor separately.
- Assuming a dimensionless Π group guarantees a useful experiment design without checking the chosen variables.
See also
FAQ
How do I convert between compound units with this tool?
Select the Unit mode, enter the compound unit expression such as L or km/h, and optionally supply a target unit. The tool expands the expression to the SI base vector, reports the scale factor k, and if a target is provided it confirms the dimensions match and gives the conversion, for example 1 L = 0.001000 m^3.
What does the equation consistency check validate?
Provide the equation and each variable’s unit. The checker expands every variable, computes dimensions for the left- and right-hand sides, and ensures additions and subtractions are between homologous dimensions. It also enforces that trig/exp/log arguments are dimensionless, so expressions such as exp(g*t) raise a flag while sin(v/v0) passes.
How are Buckingham Π groups generated?
In the Π-groups mode, list the variables and their units. The tool forms the 7×n dimension matrix, computes its null space, and returns an integer basis for the dimensionless products. For a pendulum with T, L, and g it produces Π = g^1·T^2·L^-1, matching textbook derivations.
How should I start a dimensional-analysis check?
Write every variable in consistent units first, then confirm each equation term has the same base dimensions before you interpret any scale-factor conversion. This keeps unit algebra and equation checking aligned.
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