# Dimensional analysis

By Marcus Wilson 28/09/2010

In my experimental physics class, I’ve been doing a bit of work with the students on dimensions and dimensional analysis. Most people who’ve done some physics have some intuition about it, but dimensional analysis puts it on a formal, and often useful footing.

Here’s a brief potted summary for those who don’t want to try to follow Wikipedia’s mathematically-intensive explanation.

Consider the size of your office. (Or living room, or kitchen….). Size is a vague term. What do I mean? I might mean the distance from one wall to the other wall.  This you would measure in metres (if you are metric).  Or, I might mean the area of the room. For a rectangular room, that’s the width times the length. Both those quantities are measured in metres, and so area has a unit of metres squared.  Or I could mean the volume.  Multiply the floor area by the height – and we get a value that is in metres cubed.  Note that the length, the area, and the volume are different kinds of thing. We say the length has a dimension of length (!), the area has a dimension of length squared, the volume a dimension of length cubed.

These dimensions hold true for any length or area or volume.  For example, remember the formula for the volume of a sphere:  four-thirds pi times radius cubed ?  Note that we have radius cubed here.  From the dimensions, we know it has to be radius cubed, (not squared, or to the power of 5/2 or whatever)  because a volume has a dimension of length cubed.  Somewhere in the formula there has to be a length times a length times a length.

But it’s not just lengths we have in physics.  We have time as well. For example, the velocity is how fast you are covering distance  –  i.e. a distance in a given time (e.g. 80 km an hour).  We say it has dimensions of distance per time.  Acceleration would be change in velocity in a given time, and have dimensions of distance per time squared.

And there’s more. We have masses.  A force (equal to mass times acceleration by Newton’s second law) has dimensions of mass times distance per time squared. We can also add in for good measure a dimension of electric current, and a dimension of temperature.

Now, if you are trying to work out how something depends on a lot of variables (e.g. how the drag force on a particular car depends on the density and viscosity of the air and the width and velocity of the car) the need to get dimensions correct means that the form that the equation can have is very restricted.  If I say that force is equal to some combination of density, viscosity, width, and velocity, the dimensions of this combination MUST be equal to the dimensions of force (mass times distance per time squared). This severly restricts the possible equations, and is very, very useful.

This kind of reasoning is taken to its extreme in the study of fluid flow. Physicists ably supported by engineers construct a plethora of dimensionless numbers (numbers whose dimensions cancel out – an example is the aspect ratio of a room – the length divided by the width) from quantities such as density, viscosity, velocity, temperature, acceleration due to gravity, size, heat input …  Instead of discussing the behaviour of the fluid in terms of density, velocity, etc, they discuss it in terms of these dimensionless numbers, which results in much much shorter and more manageable equations.

So, have a go at thinking in dimensions, and, when you next come across a physics equation, check for yourself that the dimensions of the left- and right-hand sides are the same. (If they’re not, something has gone wrong).

### 0 Responses to “Dimensional analysis”

• Marcus Wilson says:

Indeed. It reminds me that I was discussing earlier this year with Alison Campbell about putting in a bid to the NZ Marsden fund for studying the effect of science cartoons on the public’s understanding of science. I’m not really sure here whether I’m serious or not. Anyone interested in getting onboard?