Dimensional Analysis

Dimensional analysis is a tool used by scientists and engineers to check that their equations and calculations can possibly make sense. It cannot prove that calculations are right, but can prove that there is something wrong that needs fixing. Dimensional analysis makes use of the units of measurement: any measure of length, for example, always has units of metres, feet, fathoms, or anything else that represents distance. You can use these units to make sure that the numbers you’re putting together give you the result you want.

For example, if we add two lengths together, the units of our answer should be length. If we multiply two lengths together, the units of our answer should be length squared (area), and so forth. If this isn’t the case, that’s a very good indication that something is wrong! Similarly, dimesional analysis provides clues about the formula you need. Take the area of a circle for example. You know that the unit of area is the square metre. You also know that you only need the radius r or diameter d to define a circle. So dimensional analysis tells you that the circle’s area has to be “something” times r² or d². It doesn’t tell you which, or that the “something” has a π in it. It does, however, tell you the area cannot possibly be 2πr.

We can also use dimensional analysis the other way – we know what units we expect for our answer, but we aren’t sure what numbers we should put together to get that answer. A simple example is as follows: suppose we want to calculate the amount of CO₂ generated by driving an SUV.

Sample Calculation – CO₂ Emissions

There are some numbers which common sense tells us we’re going to need. It should be apparent that the farther the distance you drive, the greater your emissions will be if all else is equal. Let’s assume that we’re driving for 50 km. Also, common sense tells us that the fuel economy of your car will play a role, too – one would expect an army tank to emit a fair bit more than a Smart Car on a given trip. We’ll assume the fuel economy of the SUV is 0.1 L/km, and every litre of fuel burned creates 2.3 kg of CO₂ (2.3 kg/L). But that’s all the information we have – how do we use that to get the CO₂ emissions we want?

That’s where dimensional analysis comes in. Looking at the units (Eqn.1) tells us that

$\dfrac{\textnormal{CO}_2}{\textnormal{km}} = \dfrac{\textnormal{Litres of gasoline}}{\textnormal{km}} \times \dfrac{\textnormal{CO}_2 \textnormal{ emitted}}{\textnormal{Litres of gasoline}}\tag{1}$

The litres of gasoline “cancel each other out” of the equation, and thus you’re left with CO₂ per km on both sides. If you want to know the total CO₂ , you can “cancel out” the kilometres by multiplying by the distance traveled (50 km). So now all we have to do is plug in the numbers. Using units as a guide, we’ve made our own equation! When you’re doing dimensional analysis, you’ll have to get used to working with equations using units of measurement instead of numbers. See Eqn.2:

$\textnormal{Units of CO}_2 = \dfrac{\textnormal{L}}{\textnormal{km}} \dfrac{\textnormal{kg}}{\textnormal{L}} \textnormal{km}\tag{2}$

By the way, a common way to measure CO₂ emissions is by mass: grams, kilograms, tonnes (1000 kg), megatonnes (one million tonnes), etc. Looking at the equation above, can you tell what units CO₂ is being measured in?

A quick summary of the procedure

1. Consider each unit like an algebraic constant (x, y, a, etc.). Recall that x divided by x is equal to one (x/x = 1). Similarly, a unit divided by a like unit will cancel out to no units (Eqn.3). In this case:

$\tag{3}$

2. Examine what remains, and compare it with what is expected. We ended up with kg, which is what we expected (a mass unit).
3. If you can’t get the units to cancel out, then you’re probably missing something from your equation. Look for a “conversion factor” to change one unit into another. Examples include density (which changes volume into mass), or speed (which changes distance into time).

Some helpful rules

• The “x” in a^x (some number to the x^th power) has no units. After all, what does it mean to be “to the power of a metre?”
• Ratios (such as x/y) will have no units if both the numerator (x) and the denominator (y) use the same units. That’s because the units cancel out.
• Counting numbers (such as the number of apples) do not have units.

Warning!

Beware that dimensional analysis can’t be used to find absolutely everything – you need sensible data to plug in once you’ve found a formula, in order for it to make sense. That is, just because I have might find a value measured in metres per second does not mean I’ve found the speed of sound. It could be the mean in-flight velocity of a European swallow. A value like 0.3 kg/s could be how fast a factory makes copper wire, or how fast Joe the hot dog eating champion devours wieners. If you put nonsense data into a sensible equation, you will get nonsense results back out.

Summary

Dimensional analysis involves considering the units of the data used in a calculation. It can be used to check that results are reasonable, or to predict equations when the exact formula is unknown.

 

 

Revised (CEW) 2019-10-19