Dimensional analysis is the method of tracking units and dimensions through a calculation. You use it to convert units, to check that an equation makes sense and even to predict how physical quantities relate.
This guide covers all three uses, from simple conversions to the Buckingham Pi theorem, with worked examples at each step. It suits physics, chemistry and engineering courses alike.
Dimensional Analysis in Three Uses
First, convert units by multiplying by conversion factors equal to one, so unwanted units cancel. Second, check equations: every term added or equated must have the same dimensions. Third, derive relationships by matching the dimensions of the unknown quantity to those of the variables it depends on.
Dimensional analysis catches many errors, but it cannot find dimensionless constants such as 2π or ½.
Base Dimensions and Derived Quantities
The SI system builds every quantity from seven base quantities. Mechanics problems mostly need three: mass [M], length [L] and time [T].
| Quantity | SI unit | Dimensions |
|---|---|---|
| Velocity | m/s | [L T−1] |
| Acceleration | m/s2 | [L T−2] |
| Force | newton, N = kg·m/s2 | [M L T−2] |
| Energy, work | joule, J = N·m | [M L2 T−2] |
| Power | watt, W = J/s | [M L2 T−3] |
| Pressure | pascal, Pa = N/m2 | [M L−1 T−2] |
| Density | kg/m3 | [M L−3] |
| Dynamic viscosity | Pa·s | [M L−1 T−1] |
The other four SI base quantities are electric current [I], temperature [Θ], amount of substance [N] and luminous intensity [J]. Angles in radians are dimensionless.
Converting Units with Conversion Factors
A conversion factor is a fraction equal to one, such as 1000 m / 1 km. Multiplying by it changes the units without changing the quantity. Arrange each factor so the unit you want to remove sits on the opposite side of the fraction.
Example 1: speed. Convert 72 km/h to m/s.
72 km/h × (1000 m / 1 km) × (1 h / 3600 s) = 72 × 1000 / 3600 m/s = 20 m/s.
Example 2: density. Convert 1.5 g/cm3 to kg/m3.
1.5 g/cm3 × (1 kg / 1000 g) × (100 cm / 1 m)3 = 1.5 × 106 / 103 kg/m3 = 1500 kg/m3.
Example 3: flow rate. Convert 5.0 L/min to m3/s. Since 1 L = 10−3 m3: 5.0 × 10−3 m3 / 60 s = 8.3 × 10−5 m3/s.
Example 2 shows the most common trap. When a unit is squared or cubed, the conversion factor must be squared or cubed too: 1 m2 = 104 cm2 and 1 m3 = 106 cm3.
Chaining Factors in Chemistry
Chemistry courses call the same idea the factor-label method. Molar mass, density and concentration all work as conversion factors, and you can chain several in one line.
Example: grams to particles. How many formula units are in 10.0 g of sodium chloride? Molar mass of NaCl is 58.44 g/mol.
10.0 g × (1 mol / 58.44 g) × (6.022 × 1023 / 1 mol) = 0.171 mol × 6.022 × 1023 mol−1 = 1.03 × 1023 formula units.
Write each factor so the previous unit cancels, and the final unit tells you whether the chain is set up correctly. If you end with g2/mol, a factor is upside down.
A Special Case: Temperature
Conversion factors work only for scales that share a zero. Celsius and Fahrenheit have offsets, so converting a temperature needs a formula, not just a factor.
- Kelvin from Celsius: T(K) = T(°C) + 273.15.
- Fahrenheit from Celsius: T(°F) = 1.8 × T(°C) + 32.
- Temperature differences: a change of 1 K equals a change of 1 °C, and a change of 1 °C equals a change of 1.8 °F.
Use kelvin in any formula that multiplies or divides by temperature, such as the ideal gas law pV = nRT.
Checking an Equation for Consistency
The principle of dimensional homogeneity says every term added, subtracted or equated must have the same dimensions. You cannot add a length to a time, just as you cannot add apples to hours.
Check s = ut + ½at2. Left side: [L]. First term: [L T−1][T] = [L]. Second term: [L T−2][T2] = [L]. Every term is a length, so the equation is consistent. The ½ is dimensionless and does not affect the check.
Check a wrong equation, v = u + as. The term as has dimensions [L T−2][L] = [L2 T−2], but v is [L T−1]. The equation is inconsistent, so it must be wrong. (The correct relation is v2 = u2 + 2as, where every term is [L2 T−2].)
The arguments of functions such as sin, cos, exp and log must be dimensionless. In exp(−t/τ), the time t is divided by a time constant τ, so the argument has no units.
Passing the check does not prove an equation is right; it can still have a wrong constant or a missing term. Failing the check proves it is wrong.
Deriving a Relationship from Dimensions
If you know which variables a quantity depends on, dimensions alone can often give the form of the formula. Assume a product of powers and match the exponents of each dimension.
Worked example: the period of a pendulum. Suppose the period t depends on length l, mass m and gravitational acceleration g, so t = k la mb gc, where k is a dimensionless constant.
Dimensions: [T] = [L]a [M]b [L T−2]c.
Match exponents. Mass: b = 0. Time: −2c = 1, so c = −½. Length: a + c = 0, so a = ½.
Result: t = k√(l/g). The period does not depend on mass at all. A full analysis for small swings gives k = 2π, which dimensions alone cannot supply.
The Buckingham Pi Theorem
For bigger problems, the Buckingham Pi theorem organises the method. If a problem involves n variables built from k independent dimensions, it can be described by n − k dimensionless groups, called Pi groups.
Example: drag on a sphere. The drag force F depends on fluid density ρ, speed v, diameter D and viscosity μ. That is n = 5 variables built from k = 3 dimensions (M, L, T), so there are 5 − 3 = 2 groups.
One choice is the drag group F / (ρv2D2) and the Reynolds number Re = ρvD/μ. So F / (ρv2D2) = f(Re): instead of testing five variables, you only need one curve.
This is why wind tunnel and towing tank tests work. A small model matched to the full-size object's Reynolds number, or whichever group dominates, behaves in the same dimensionless way.
Common Mistakes
| Mistake | Fix |
|---|---|
| Not cubing the factor for volumes | (100 cm / 1 m)3 = 106 cm3/m3 |
| Dropping units part-way through | Write units on every line of working |
| Celsius in pV = nRT | Convert to kelvin first |
| Treating a passed check as proof | It only shows the equation could be right |
| Mixing unit systems | Convert everything to SI before substituting |
Mixed units have real costs. NASA's Mars Climate Orbiter was lost in 1999 partly because one team's software produced thruster data in pound-force seconds while another expected newton-seconds.
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Frequently Asked Questions
Converting between units, checking that equations are consistent, deriving the form of physical relationships and designing scale-model experiments.
A dimension is the type of quantity, such as length [L]. A unit is a particular measure of it, such as metres or feet. Many units share one dimension.
No. Dimensionless constants are invisible to dimensional analysis. You need theory or experiment to find them.
These functions are defined by power series that add terms with different powers of the argument. Those terms could only be added if the argument has no dimensions.
The number of variables minus the number of independent dimensions they involve, by the Buckingham Pi theorem.
The factor-label method, common in chemistry, is the unit-conversion part of dimensional analysis. The broader method also covers equation checks and derivations.
Radians are a named unit, but angles are dimensionless because a radian is a ratio of arc length to radius.