Dimensions
Strip a quantity down to powers of mass, length and time.
Lesson at a glanceShow the key points
- Every derived quantity can be written as a product of powers of base quantities; the powers are its dimensions.
- Symbols: M (mass), L (length), T (time), I (current), K (temperature), mol, cd; square brackets mean 'dimensions of'.
- Force = mass × acceleration, so [F] = MLT⁻²: dimension 1 in mass, 1 in length, −2 in time.
- Pure numbers (½, 2π) and ratios of like quantities are dimensionless; only the kind of quantity matters, not its size.
- Examples: velocity LT⁻¹, acceleration LT⁻², energy/work ML²T⁻², power ML²T⁻³, pressure ML⁻¹T⁻², momentum MLT⁻¹, density ML⁻³.
Forget the numbers for a moment and ask only what kind of thing a quantity is. A speed is a length per time. A force is a mass times a length per time squared. That skeleton is its dimensions.
Building quantities from base quantities
Every physical quantity can be built from the base quantities by multiplying and dividing. Follow force back to its roots:
The power to which each base quantity appears is called the dimension of the quantity in that base. Force has dimension 1 in mass, 1 in length and −2 in time, and zero in every other base quantity.
Only the kind of quantity matters, never its size. Initial velocity, final velocity, change in velocity and average velocity are all simply length/time.
Dimensional formulas
Use one-letter symbols: M for mass, L for length, T for time, I for electric current; temperature, amount of substance and luminous intensity use K, mol and cd. Square brackets mean “the dimensions of”:
Find the dimensional formulas of (a) pressure, P = F/A, (b) power, P = W/t, (c) the gravitational constant G in F = Gm₁m₂/r².
| Quantity | Built as | Dimensional formula |
|---|---|---|
| Velocity | length ÷ time | |
| Acceleration | velocity ÷ time | |
| Momentum | mass × velocity | |
| Work, energy | force × distance | |
| Power | work ÷ time | |
| Pressure | force ÷ area | |
| Density | mass ÷ volume | |
| Frequency | 1 ÷ time |
- 1The dimensions of a quantity are the powers of the base quantities it is built from.
- 2Written as a dimensional formula in square brackets: .
- 3Magnitude is irrelevant; pure numbers and ratios of like quantities are dimensionless.
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