At some point in the introductory physics class, the students look at the idea of center of mass. The usual explanation is that when you are dealing with a rigid object, you need to know what forces are acting on the object and where these forces are acting. The difficult force is the gravitational force. This actually pulls on all parts of a rigid object. However, it turns out that if we model the gravitational force as though it was just acting at one point, you would get the same result as if you assumed it pulled on all parts. We usually call this point the center of mass - but technically, it is the center of gravity. Here is an older post where I show where this center of gravity comes from.

But most textbooks just say that if you have some individual masses at different locations, you can find both the x and y center of mass with the following calculation.

Maybe this is easier to understand with an example. Suppose I have three masses along the x-axis. Each mass has an x-coordinate and a mass. The location of the x-center of mass would then be:

See. That's not so difficult. If the masses had y-coordinates too, you would do the same thing in the y-direction. If there were 20 masses instead of 3, you would just have more terms. If you have a continuous mass, you could break it into many, many small pieces and either integrate or use a numerical method to determine the center of mass.

A Center of Mass Example ————————

It turns out that there aren't very many cases in the introductory algebra-based course where you would find the center of mass of discrete point masses. Yes, there are a couple of cases but they are either boring or contrived.

If we are going to do useless examples, how about one that is at least fun? What is the center of population of the Southeastern Conference (SEC)? What does this even mean? What if I look at all the universities in the SEC as well as the student population for these institutions? I could use this population instead of the mass to find the point on the Earth that is the center of population.

Let's do it. First, there are 14 institutions in the SEC. I am going to use the student population values listed in Wikipedia - because truthiness.

Here is the plan. Use Google Maps and Tracker Video Analysis to get x and y coordinates of the different schools. Yes, I can use Tracker Video Analysis even thought I don't have a video. Here is the map I will start with.

If you use the "classic" version of Google Maps, you can still mark two points and get the distance between these two points. Really, I just picked two random locations that were far apart. Now, when I pull this image into Tracker, I can scale the map based on this distance. One of the important aspects of finding the center of mass is to first pick an origin. It doesn't really matter exactly where this origin is located, but once you have it your location of the center of mass will be relative to this position. I don't know why, but I chose the origin to be in New Orleans.

If you look at the image from Google Maps closely, you might notice something interesting. The red line is the path on Google maps between the two locations I picked and the blue line is the calibration tape from Tracker Video. The two lines are not the same. The line from Google is curved to represent the shortest distance between these two points. However, since the map is flattened out this short line appears curved. Tracker video, on the other hand, doesn't know this is supposed to be the Earth so it just draws a straight line. I just thought it was cool that Google takes into account the curvature of the Earth for these measurements.

Once I have an origin, I can just mark different points on the image to get the x-y coordinates of that location. Technically, these values will be off a little bit because of the flat image, but it will be close enough for this example.

Now I just need to calculate and plot this SEC center. Here it is.

Base map image from

I included a bubble plot showing both the individual institutions and the center of the SEC. The size of the bubble is proportional to the enrollment for that university. So, where is the center of population for the SEC? It looks to be just North of Tuscaloosa.

I wonder if the SEC championship game should be held in Birmingham Alabama instead of Atlanta Georgia?

Homework ——–

There are a whole bunch of great questions here. Also, this stuff isn't too difficult to calculate.