So many pieces!
Finally, finally we have arrived at the point all of these posts have been building toward:
In physics, we only occasionally care about “area”. But we often care about adding up an infinite number of tiny little things. And when we do, if we think of them as representing areas, we can use our Magic Theorem to add them.
I want to demonstrate this using as little physics as possible. So this next example may seem a bit contrived. You will see more significant examples as our year goes on (perhaps more in E & M than in Mechanics). But this is a good one to begin with:
A beam is 2 meters long. It has a density that varies along its length. As a function of distance from the left end, the linear density (in kg/m) is given by the function:
Find the mass of the beam.
Before we jump in, let me say a little about what we are hoping to do.
We have a quantity that varies continuously: in this case, it’s the density. (Also, please do not be intimidated by the fact that we are using the Greek letter, lambda, to represent the density function. I think we do this because it is a LINEAR density (mass/length) and so we want a letter that begins with L.)
If we chop our beam into short pieces, we can pretend the density is constant for a given piece. Really, density does vary but if the piece is small, density doesn’t change much. And we are eventually going to make the pieces reeeeally small. This is pretty much the same thing we did last time, when we pretended velocity was constant over a time interval.
We then multiply the length of each piece by its density to find the mass of that piece and then we add them up.
So here’s the beam, nicely chopped:
We focus on one individual piece, say that blue one, and find its mass:
And we add up all those little masses:
As we have before, we can improve our estimate by chopping into more pieces – or even better, an infinite number of pieces:
Now let’s really examine that expression. It sure looks a lot like a Riemann sum! It looks like you are getting ready to find the area under the curve: f(x)=1+x-x2/2 by breaking it into rectangles. And for that we can use the Magic Theorem.
We can find the anti-derivative ourselves or use Wolfram/TI/whatever to get:
And then see how much it changes:
We are done! And yes, that is also the area under that graph, but that’s not really the point. We were not interested in finding an area. We just wanted to know the mass. But as we have just seen, the same method that lets us find areas also helps us to add up the little pieces!
Well, that’s all I wanted to say about the Fundamental Theorem and how it is used in physics. If you have stayed with me through this entire 5-part journey, thank you so much for your persistence. I find this to be pretty challenging stuff. I will be happy to help you through it again when we encounter this during our year in AP Physics. I hope the time you spent working through these posts makes it easier for you when you see it again.
See you soon!