Dimensional analysis

Dimensional analysis is a useful tool used in engineering and science to study physical quantities like length, mass, and time. It helps us understand how these quantities relate to each other in mathematical expressions. This idea was first introduced by Joseph Fourier in 1822.

Physical quantities that have the same dimension, like length or time, can be compared directly even if they use different units, such as metres and feet or seconds and years. However, quantities with different dimensions, like length and mass, cannot be compared — for example, it doesn’t make sense to ask if a gram is larger than an hour.

One important rule in physics is that any meaningful equation must have the same dimensions on both sides. This helps scientists check if their equations make sense and guides them when they are trying to create new equations to describe how a system works.

Formulation

The Buckingham π theorem helps us understand how physical equations work. It says that any equation with many parts can be rewritten using fewer, special values called dimensionless parameters.

Dimensional analysis looks at the basic parts of measurements like length, mass, and time. For example, mass is a basic part, while a kilogram is a specific way to measure mass. Different units, like meters or feet for length, don't change the basic idea of length.

The SI system uses seven basic measurements: time, length, mass, electric current, temperature, amount of substance, and light brightness. These help us understand and compare many different kinds of physical quantities.

Concrete numbers and base units

Many measurements in science and engineering use a number and a unit together. For example, speed can be shown as how far something goes in a certain time, like 60 kilometres per hour. We use division to show "per", such as km/h. We can also multiply units, use powers like m² for area, or combine these methods.

Some units are chosen as base units because they cannot be made from other units. For example, we usually pick units for length and time as base units. Other units, like volume, can be made from these base units, such as using metres to show cubic metres (m³).

Sometimes unit names hide that they are made from other units. For example, a newton (N) is a unit for force. It is defined as 1 kilogram multiplied by metre per second squared (1 kg⋅m⋅s⁻²).

Percentages are special because they do not have units. They show ratios of two things with the same units, like comparing lengths or times.

When we take derivatives in math, we divide the unit by the unit of what we are changing. For example, velocity (how fast something moves) has units of length divided by time.

Integrals in math add the unit of what we are integrating. For example, work, which is force multiplied by distance, has units that combine force and distance.

In economics, we talk about stocks (like money) and flows (like money per year). Sometimes, we compare stocks and flows as percentages, even though they have different units.

Dimensional homogeneity (commensurability)

Main article: Apples and oranges

Further information: Kind of quantity

The most basic rule in studying how different measurements relate to each other is that only things that are the same kind can be compared or added together. For example, you can compare 1 hour to 2 hours, or 1 kilometre to 2 kilometres, because they are both measurements of time or distance. But it doesn’t make sense to compare an hour to a kilometre because one measures time and the other measures distance.

However, we can still work with measurements that are different kinds by using division or multiplication. For instance, if a car travels 100 kilometres in 2 hours, we can divide these to find the speed: 100 kilometres ÷ 2 hours = 50 kilometres per hour. This works because we’re turning two different kinds of measurements into one that tells us about speed.

When checking if a science formula makes sense, all the parts of the formula need to be the same kind. For example, adding the weight of a person and the weight of a rat is okay because both are weights. But adding a person’s weight to their height doesn’t make sense — one is a weight, and the other is a length!

Conversion factor

Main article: Conversion factor

In dimensional analysis, a special number called a conversion factor helps change one measuring unit into another without changing the actual amount. For example, both kPa and bar are ways to measure pressure, and 100 kPa equals 1 bar. Using math rules, we can divide both sides of this to get 100 kPa divided by 1 bar equals 1. This number, 100 kPa / 1 bar, can be used to switch from bars to kPa by multiplying it with the amount we want to change. So, if we have 5 bar and multiply by 100 kPa / 1 bar, we get 500 kPa because the "bar" units cancel out, showing that 5 bar is the same as 500 kPa.

Applications

Dimensional analysis is often used in physics and chemistry, but it can also be helpful in other areas.

In mathematics, dimensional analysis helps us understand shapes. For example, when calculating the volume of a ball in many dimensions, we can see how it changes with size by looking at the power of the measurement.

In finance, economics, and accounting, dimensional analysis helps us understand the difference between things that stay the same (like money) and things that change over time (like income). It also helps us interpret important financial ratios, such as the P/E ratio, which tells us how many years it would take to earn back the price paid for a stock.

In fluid mechanics, dimensional analysis is used to create special numbers that describe how fluids behave. These numbers, like the Reynolds number, help scientists and engineers understand and predict fluid flow without doing complex experiments for every situation.

History

The history of dimensional analysis is interesting and full of discovery. It all started with François Daviet, a student of Joseph-Louis Lagrange, who wrote about it in 1799. Later, Joseph Fourier made big steps in 1822, showing that important science rules should work no matter what units we use.

Great scientists like James Clerk Maxwell helped shape how we use dimensional analysis today. They focused on basic units like mass, length, and time, and used them to understand more complex ideas. This way of thinking helped explain many natural wonders, like why the sky looks blue.

Examples

A simple example: period of a harmonic oscillator

We can use dimensional analysis to find the period of a mass attached to a spring. The period depends on the mass, the spring constant, and gravity. By looking at the units of these quantities, we can see that gravity does not affect the period. This means the period would be the same on Earth or the Moon. The analysis shows us that we only need the mass and spring constant to find the period.

A more complex example: energy of a vibrating wire

Dimensional analysis and numerical experiments for a rotating disc

For a vibrating wire, we can use dimensional analysis to understand how its energy depends on different factors like length, amplitude, density, and tension. The analysis shows that the energy depends on the tension and amplitude, but not on the wire’s density. This helps us simplify experiments because we know some factors do not matter.

A third example: demand versus capacity for a rotating disc

When studying a rotating disc, engineers can use dimensional analysis to understand how different factors like density, size, and rotation speed affect the stress in the disc. By grouping these factors into dimensionless groups, they can create charts that help design and assess rotating discs. This makes it easier to see how changes in one factor affect the overall performance of the disc.

Properties

The study of how we describe physical things using measurements is called dimensional analysis. It helps us understand how different measurements like length, mass, and time relate to each other in calculations.

This idea was first introduced by Joseph Fourier in 1822. When we have measurements that can be compared directly, like meters and feet or grams and pounds, they share the same type of measurement. This means we can still compare them even if they use different units.

Dimensionless concepts

Constants

Main article: Dimensionless quantity

Some special numbers, called constants, appear in science when we study certain problems. These numbers help us understand how nature works. For example, in fluid flow or spring behavior, we might find a constant called C or κ. Even though dimensional analysis doesn’t tell us much about these numbers, they are often close to 1. This helps scientists make quick guesses about how things will behave, plan experiments, or decide what is important to study.

Formalisms

Interestingly, dimensional analysis can still be helpful even when all the pieces of a theory have no units. For example, in models that study how materials change at a tiny level, like the Ising model, we can look at how things behave when they get very close to a special point. In these cases, a special distance called the correlation length, χ, grows larger and larger. This distance helps us understand important changes, and we can use dimensional analysis to guess how certain properties change with this distance.

Some scientists, like Michael J. Duff, suggest that the rules of physics might actually have no units at all. They believe that giving units to things like length, time, and mass is just a habit from older science. According to this view, important constants like c, ħ, and G act like bridges to connect these ideas. By thinking of these constants as having no units, we can better understand how physics works in extreme situations.

Dimensional equivalences

Here are some common expressions in physics that relate to energy, momentum, and force. The tables below show how these quantities connect through their dimensions.

Energy, E (T⁻²L²M)	Expression	Nomenclature
Mechanical	W = F d {\displaystyle W=Fd}	W = work, F = force, d = distance
	S / t ≡ P t {\displaystyle S/t\equiv Pt}	S = action, t = time, P = power
	m v 2 ≡ p v ≡ p 2 / m {\displaystyle mv^{2}\equiv pv\equiv p^{2}/m}	m = mass, v = velocity, p = momentum
	I ω 2 ≡ L ω ≡ L 2 / I {\displaystyle I\omega ^{2}\equiv L\omega \equiv L^{2}/I}	L = angular momentum, I = moment of inertia, ω = angular velocity
Ideal gases	p V ≡ N T {\displaystyle pV\equiv NT}	p = pressure, V = volume, T = temperature, N = amount of substance
Waves	A I t ≡ A S t {\displaystyle AIt\equiv ASt}	A = area of wave front, I = wave intensity, t = time, S = Poynting vector
Electromagnetic	q ϕ {\displaystyle q\phi }	q = electric charge, ϕ = electric potential (for changes this is voltage)
	ε E 2 V ≡ B 2 V / μ {\displaystyle \varepsilon E^{2}V\equiv B^{2}V/\mu }	E = electric field, B = magnetic field, ε = permittivity, μ = permeability, V = 3d volume
	p E ≡ m B ≡ I A B {\displaystyle pE\equiv mB\equiv IAB}	p = electric dipole moment, m = magnetic moment, A = area (bounded by a current loop), I = electric current in loop

Momentum, p (T⁻¹LM)	Expression	Nomenclature
Mechanical	m v ≡ F t {\displaystyle mv\equiv Ft}	m = mass, v = velocity, F = force, t = time
Mechanical	S / r ≡ L / r {\displaystyle S/r\equiv L/r}	S = action, L = angular momentum, r = displacement
Thermal	m ⟨ v 2 ⟩ {\displaystyle m{\sqrt {\left\langle v^{2}\right\rangle }}}	⟨ v 2 ⟩ {\displaystyle {\sqrt {\left\langle v^{2}\right\rangle }}} = root mean square velocity, m = mass (of a molecule)
Waves	ρ V v {\displaystyle \rho Vv}	ρ = density, V = volume, v = phase velocity
Electromagnetic	q A {\displaystyle qA}	A = magnetic vector potential

Force, F (T⁻²LM)	Expression	Nomenclature
Mechanical	m a ≡ p / t {\displaystyle ma\equiv p/t}	m = mass, a = acceleration
Thermal	T δ S / δ r {\displaystyle T\delta S/\delta r}	S = entropy, T = temperature, r = displacement (see entropic force)
Electromagnetic	E q ≡ B q v {\displaystyle Eq\equiv Bqv}	E = electric field, B = magnetic field, v = velocity, q = charge

Programming languages

Dimensional analysis can be used in computer programming to make sure calculations are correct. This idea has been studied since 1977. Different programming languages like Ada, C++, Standard ML, F#, Haskell, OCaml, Rust, and Fortran have ways to handle units and measurements in code.

A tool called Mathematica has special functions to work with measurements. It can change equations to remove units, find the units of a measurement, and see which groups of measurements have the same units. These tools help programmers and scientists make sure their calculations make sense.

Geometry: position vs. displacement

Affine quantities

Dates and positions are not the same as durations and displacements. Dates are labels, while durations show how much time has passed. You can add two durations to get a new duration, and you can add a duration to a date to get a new date. But you cannot add two dates together — that doesn’t make sense.

Vectors can be added together to make new vectors. Affine quantities, like dates, can only be subtracted to give relative differences, which are vectors. To represent affine quantities like dates, you need a reference point and a coordinate system.

Orientation and frame of reference

In multi-dimensional space, a displacement isn’t just a length; it also has a direction. To compare or combine quantities in space, you need a frame of reference.

Huntley's extensions

Huntley suggested looking at the components of vectors separately. For example, instead of just length, you can have length in the x-direction, y-direction, and so on. He also suggested that mass as a measure of matter can be different from mass as a measure of inertia.

Siano's extension: orientational analysis

Angles are usually seen as dimensionless, but Siano suggested using orientational symbols to better understand them. These symbols help show the direction of angles and can give more information about physical problems. For example, in projectile motion, this method can show that certain functions of the angle make more sense than others.

Symbol	Variable	Dimension
m ˙ {\displaystyle {\dot {m}}}	mass flow rate	T⁻¹M
p x {\displaystyle p_{\text{x}}}	pressure gradient along the pipe	T⁻²L⁻²M
ρ	density	L⁻³M
η	dynamic fluid viscosity	T⁻¹L⁻¹M
r	radius of the pipe	L

	1 0 {\displaystyle \mathbf {1_{0}} }	1 x {\displaystyle \mathbf {1_{\text{x}}} }	1 y {\displaystyle \mathbf {1_{\text{y}}} }	1 z {\displaystyle \mathbf {1_{\text{z}}} }
1 0 {\displaystyle \mathbf {1_{0}} }	1 0 {\displaystyle 1_{0}}	1 x {\displaystyle 1_{\text{x}}}	1 y {\displaystyle 1_{\text{y}}}	1 z {\displaystyle 1_{\text{z}}}
1 x {\displaystyle \mathbf {1_{\text{x}}} }	1 x {\displaystyle 1_{\text{x}}}	1 0 {\displaystyle 1_{0}}	1 z {\displaystyle 1_{\text{z}}}	1 y {\displaystyle 1_{\text{y}}}
1 y {\displaystyle \mathbf {1_{\text{y}}} }	1 y {\displaystyle 1_{\text{y}}}	1 z {\displaystyle 1_{\text{z}}}	1 0 {\displaystyle 1_{0}}	1 x {\displaystyle 1_{\text{x}}}
1 z {\displaystyle \mathbf {1_{\text{z}}} }	1 z {\displaystyle 1_{\text{z}}}	1 y {\displaystyle 1_{\text{y}}}	1 x {\displaystyle 1_{\text{x}}}	1 0 {\displaystyle 1_{0}}