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Dimensions of a quantity
Intermediate
Expression of a quantity in powers of base dimensions M, L, T.
[F] = M L T⁻²The dimension of a quantity states how it is built from base dimensions: mass [M], length [L], time [T] and others. It ignores units and numbers. Velocity is [L T⁻¹], acceleration [L T⁻²], force [M L T⁻²], energy [M L² T⁻²].
Pressure = force/area has dimensions M L T⁻² / L² = M L⁻¹ T⁻².
Dimensions are not the same as units: metres and feet differ in unit but share the dimension [L].
In more detail
The dimensions of a quantity describe its base-unit makeup, using symbols such as M for mass, L for length and T for time. Speed has dimensions L T to the power minus 1, while force has M L T to the power minus 2.
Real-world uses
- Dimensional analysis confirms that a formula for pendulum period contains only time, not mass.
- Engineers check that a simulation's equations use consistent dimensions before running it.
- Checking that a formula for power has the dimensions of energy per time catches many errors in engineering calculations before any numbers are entered.
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Frequently asked questions
What are the dimensions of energy?
M L² T⁻², the same as work, since energy is force times distance.
Are unit and dimension the same?
No. Metres and feet are different units but both have the dimension of length.
What are the dimensions of force?
Force has dimensions M L T⁻², because force equals mass times acceleration.
Related topics in Measurement & Math Tools
Error in product or quotient · Error in powers · Mean and standard deviation · Dimensional homogeneity · Checking equations by dimensions · Deriving formulas by dimensions