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Checking equations by dimensions
Intermediate
Dimensional analysis can reject (not prove) a formula.
To test a formula, write the dimensions of each side. If they differ, the formula is definitely wrong. If they match, it may still be wrong by a numerical factor or have the wrong structure.
Is T = 2π√(g/L) valid? √(g/L) has dimension T⁻¹, not T, so no. The correct one is √(L/g).
Matching dimensions is necessary but not sufficient. v = ½ at passes, and so does v = 5at.
In more detail
To check an equation, replace each quantity with its dimensions and simplify both sides. If the two sides differ, the equation has an error in its structure, even if the numbers look plausible.
Real-world uses
- Checking the formula v = sqrt(2 g h) by dimensions confirms the speed has units of metres per second.
- Students use this check during exams to spot a wrong power of a variable.
- A student proposes a speed formula v = d + t and sees immediately that metres cannot be added to seconds, so the proposal is rejected.
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Frequently asked questions
What do I do if the dimensions do not match?
Look for a missing or extra variable, or a wrong power, and rederive the formula from its physical basis.
Can I check trigonometric functions this way?
Only their arguments need to be dimensionless. Angles are treated as dimensionless, so sin θ is a pure number.
Can a trigonometric function have dimensions?
No. Functions such as sine and cosine take dimensionless arguments, so an angle is treated as a pure number.
Related topics in Measurement & Math Tools
Mean and standard deviation · Dimensions of a quantity · Dimensional homogeneity · Deriving formulas by dimensions · Limitations of dimensional analysis · Dimensionless numbers