Six Months in Copenhagen

On the most basic level, "units" are what separate physics from mathematics. They give physical meaning to numbers. Understanding the connection between units, numbers and reality is the fundamental step in doing any physical computation. This understanding is not as basic as it may sound. Stepping through the parts of a problem, the units of different variables may vary. You might have to convert units between systems (metric vs. English), order of magnitude (N vs kN), or even based on notation (MPa vs. kN/m^2).

The connection between units and calculation is not always obvious to beginning engineering students. However, by being aware of the units of the variables and answers, many mistakes can be made obvious. The first check of a solution should be whether the units of your result match the desired units. For example, when computing the moment in a cantilever (Moment = Force * Distance), the result must have units of force * length (i.e. lb*ft). Any additional unit terms should tip you off to a mistake somewhere.

Every couple of years, a big push is made to convert measurement in the U.S. to metric units.  I can't understand why there is always so much resistance; I have actually enjoyed using metric units here in Denmark. The metric system makes a lot of sense because conversions are usually a function of the number 10. Instead of remembering to convert 5,280 ft to 1 mile; 1,000 m is 1 km. My only problem was visualizing the physical weight, size or volume - but I'm sure that would come with practice.

One day I was having some problems with my calculations. I was putting together a spreadsheet on floor vibration. Through the various calculation steps, I needed to convert by multiples of 1000 to make sure that my units were complimentary. When I got to the end, I found that my answer was off - obvious because the magnitude was significant. I lost track of my unit conversions somewhere. I had to map out the multiple calculation steps by hand to find the error.

As it turns out, I was making two mistakes. The first involved converting kN to kg (Force to mass). kN = 1000*kg*m/s^2. The "m/s^2" part relates to gravitational acceleration. In the English system, we seldom use the unit for mass (slugs); we change the equations to work in units of force (lb). My second mistake was forgetting to convert a fundamental frequency from the radial system by a factor of 2*pi.  I was used to using equations that automatically included the conversion.

Eventually, I problem solved my way through the worksheet and arrived at something that actually worked out.  By keeping track of the units, I was able to trace my errors.  This is a simple trick that engineers use all the time.