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Dimensional homogeneity
Intermediate
Every term in a valid equation has the same dimensions.
Only quantities with the same dimensions can be added, subtracted or equated. Every term in a physical equation must therefore have identical dimensions. Arguments of sine, exponential and log must be dimensionless.
In s = ut + ½at², each term is [L]: ut is (L/T)·T, and at² is (L/T²)·T².
The equation x = sin(t) is dimensionally invalid because t has units; it must be sin(ωt) with ω in 1/time.
In more detail
An equation is dimensionally homogeneous when every term on both sides has the same dimensions. This is a quick test for catching errors in formulas.
Real-world uses
- If a kinetic energy formula produces a result in kg m/s, you know the formula is wrong.
- Physics exam answers are checked this way to catch sign and power mistakes.
- When adding a velocity to a distance, the equation fails the check immediately, so the mistake is found without any measurements.
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Frequently asked questions
Can a dimensionally correct equation still be wrong?
Yes. Dimensions cannot detect missing numerical factors such as the 1/2 in kinetic energy, or missing terms.
Do constants like pi need dimensions?
No. Pure numbers are dimensionless and do not affect the dimension check.
Is a dimensionally consistent equation always correct?
No. It can still have a wrong numerical factor, such as a missing 1/2, so the check can reject a formula but cannot confirm it.
Related topics in Measurement & Math Tools
Error in powers · Mean and standard deviation · Dimensions of a quantity · Checking equations by dimensions · Deriving formulas by dimensions · Limitations of dimensional analysis