EN3: Introduction to Engineering and Statics
Division of Engineering
Brown University
2.3 Dimensional Analysis
We noted in the preceding section that we completely quantify the universe using only 6 basic units of measurement. In fact, in this course, only 3 are going to be of interest to us – mass, length and time; in statics problems we don’t even care about time.
This has a deep and vitally important consequence – leading to the concept of Dimensional Analysis. Unfortunately the concept is so deep that I don’t know how best to teach it, even though I make use of it daily. This suggests that I don’t really understand the concept myself. Still, here goes.
At its most basic level, dimensional analysis gives us a way to check a formula. The idea here is that if our models of the universe make sense, any formula must be independent of our choice of basic units. This is possible only if the basic units on the left and right hand sides of a formula agree.
For example consider the formula given earlier for the period of a swinging pendulum
The left hand side clearly has basic units (or dimensions) of time – how about the right hand side? To see, express everything in the formula in terms of basic units:
(and 
is said to be dimensionless
– it has no units, it’s just a number). Plugging into the formula
so the right hand side also has dimensions of time, and everything is consistent under a change of units.
This is a handy trick to check all formulas. For example, in the expression for kinetic energy K of a particle of mass m moving with speed v
recall that energy (force * distance) has dimensions 
, while the right hand side has dimensions 
so this checks out too.
Exercises: check out the dimensions of the following formulas:
1. Kinetic energy of a disk of radius a and mass m spinning with angular
velocity 
(radians per sec) :
2. Period of oscillation T of a mass m connected to a spring of stiffness k
3. Lift force 
generated by an airfoil with area A moving at speed v through air of
density 
Here, 
is the lift
coefficient of the airfoil, which depends on its angle of attack and its shape, but
nothing else.
Any time you derive a formula, you should always check to make sure it is dimensionally consistent. It is amazing how many errors you can catch this way.
Advanced Topic: This is all simple enough, but dimensional analysis is a deeper concept than just a check on formulas. It also tells us how the world scales. This is where things get trickier. I will do my best to explain how this works, because it’s a very powerful concept. But don’t worry if you don’t get it right away – many grad students have a hard time with it. You will revisit the idea in later fluid mechanics courses in any case.
First, note that, if dimensions of left and right hand sides of a formula always agree, it must always be possible to rearrange the formula so that both left and right hand sides are dimensionless. For example, for the pendulum example:
Now both left and right hand sides of the formula have no dimensions – they are just numbers.
Similarly, for the airfoil example
again, in the new formula left and right hand sides are both dimensionless.
So what? Well, this is a general principle. In any engineering problem, we want to predict, or measure, some quantity – time, length, speed, etc, which will be a function of variables and parameters in the problem – density, mass, length, applied forces, etc. A general functional relationship would look like this
Here F[] is some function – we don’t know what it is, we might determine it experimentally or by doing computer simulations. But dimensional analysis tells us that we can re-write this functional relationship in dimensionless form:
We can arrange our formula in this form even if we don’t know how to find the function F. All we need to know are the dimensions (basic units) of the quantity we are trying to solve for, and the dimensions of the various parameters.
This is a bit mysterious and confusing so let’s illustrate it with a specific example. Suppose we want to find the period of oscillation of a pendulum. We have no idea how to do the calculations – instead, we are going to do a bunch of experiments or computer simulations to find T.
By thinking about the problem, we might guess that
where m is the mass of the pendulum, L is its length, and g is the acceleration due to gravity. The function F is unknown, and we have to find it experimentally or using computations.
This would be a real chore. Even if we only do experiments (or simulations) for 3 different masses, 3 different values of g and three different lengths, we have to do everything 27 times to get an idea of the function F.
But let’s write the functional relationship in dimensionless form. To do this, we just need to combine our variables to create dimensionless groups. On the left hand side we have time, T, so we need to multiply and/or divide T by combinations of variables from the right hand side until the left hand side is dimensionless. In this case, the only possibility is
We need to do the same on the right hand side – i.e. we need to combine m, g, and L to create a dimensionless group (I don’t want, or need to include time as that’s supposed to stay on the left hand side of the formula). But this can’t be done – there is no combination of m, g and L which is dimensionless! So this means the right hand side can only be a constant, independent of any of our variables. So we conclude that
This is amazing! Now, all we need to do is one, single experiment or computer simulation to determine the constant, and we can immediately predict how the period varies with pendulum length, acceleration due to gravity and pendulum mass.
Let’s make things a bit more complicated. Suppose we guess that
where 
is the angle of
swing. In this case, we note that is dimensionless, so this time we can form a dimensionless group to
stay on the right hand side of our expression. In dimensionless form, our relationship
becomes
where f is an unknown function. Now we’d have to do a bunch of experiments
for various values of , but we
don’t need to vary anything else. Suppose we want to account for the effects of air
resistance. We might guess that
where 
is the density of air
and a is the radius of the pendulum. Then,
There’s one last issue here – how do we know how many dimensionless groups to
form? For example, in the preceding example 
is also a dimensionless group – shouldn’t we include that too?
What about 
? Fortunately, the answer is no. The dimensionless groups must all be independent
– that is to say, no dimensionless group is allowed to be a product (or division) of
any other set of dimensionless groups. In this case
in other words, the proposed additional dimensionless groups are functions of existing ones, making the new groups redundant. So, in any problem, you simply go on forming dimensionless groups until you can’t find any more independent ones.
Exercise:
A projectile of mass m is launched at speed V at angle 
. It travels for a distance d
and is airborne for time t. Write the functional relationships
in dimensionless form. Add variables that might account for air resistance, and reduce the results to dimensionless form.