Thursday, January 30, 2014

Physicists do it all the time

Physicists do it all the time, and so should geologists and engineers.

I'm talking about dimensional analysis. As a student in a physics department (which I was, for my undergrad degree in geophysics), it is required as part of "showing your work" but it also helps you figure out which formulas and equations to use if you're stuck.

Dimensional analysis is a tool to keep track of the units when you are doing calculations with physical properties. If you have an equation that involves measured values, the units on both sides of the equation must add up.  If you manipulating an equation in any way, such as substituting in other equations, or rearranging it to solve for an important variable, you should do the dimensional analysis before you start plugging in your numbers.

If you are working with Darcy's Law, or Newton's Law of Universal Gravitation, or any other number of equations that involve measurements of physical parameters, the first step to making sure your work is correct should always be dimensional analysis.  Do this on a piece of paper, before you start plugging things in to a spreadsheet or script/code.

Here's an example using Newton's Law of Universal Gravitation, for calculating the gravitational force between two bodies:

1. Create a "legend" of the variables in your equation, defining them and the units they are measured in:
  • Fg is the gravitational force (in Newtons)
  • m1 and m2 are masses of the two bodies (in kilograms)
  • d is the distance between the two bodies (in meters)
  • and G is the gravitational constant: 

2. Plug the units into the equation instead of the variables:

3. We can rearrange this a little bit, and cancel things out, leaving Newtons on both sides of the equation:

As long as the values you have are input in Newtons, kilograms, and meters, you will correctly calculate the gravitational force, in Newtons, between two objects.

Notice that there are no values in the dimensional analysis (steps 2-3). Dimensional analysis should never include values. 

Last semester, my students had to do two labs where they made some lab measurements and then did some calculations using those measurements in Excel. They struggled with this concept a lot in the first one.  A few weeks later, when it was time for the second lab like this, I included teaching them to do dimensional analysis before plugging things into Excel. It made a world of a difference in how well they did with the rest of the lab.