Application of mathematics to the real world

When we attempt to build a mathematical model of some aspect of the real world, we must proceed with extreme caution. In other words, it can be done wrong. Let’s look at an example of how it can be done correctly.

The stretch of a spring

Let’s start with a simple example. You have a loosely coiled steel spring and you are interested in how much a certain weight will stretch the spring. (Why? Maybe you want to build a simple fish scale and you don’t want to just go out and buy one from the local sporting goods store.)

So let’s set up an experiment: Hook one end of the spring to the ceiling. Then hook various weights to the dangling end of the spring and measure how much the spring has stretched. Look at the following illustration to see how the experiment might look.

The figure shows the spring in green, a weight object connected to the bottom end in gray, a ruler over to the left in yellow and a red pointer connected to the bottom end of the spring that points at a spot on the ruler. The weight object conveniently displays its weight in grams. The ruler measures centimeters. The figure initially shows a weight of zero grams and the red pointer points to about 250 centimeters on the ruler. The pointer in the weight slider, labeled “grams” along the right side in the figure can be moved up or down to simulate changing the number of grams of the weight object. As you change the weight, the spring stretches to a new length and the red pointer points to a new number on the ruler.

Ok, now what? Let’s take some measurements from our experimental apparatus. Do this:

• Move the weight slider to some position.
• Record the weight shown on the weight object and next to it, in the length column, the value on the ruler pointed to by the red arrow. We have provided a convenient worksheet below for you to record your observations.
• Record a couple more observations on the worksheet.
• Now pick a number of your choosing somewhere between 0 and 300 but different from the three weights you have already recorded. Record this number in an empty row in the weight column.
• Try to guess what number should be recorded in the length column for this weight.

Can we do better than just trying to guess what the length value will be for any weight we choose? We want to produce a rule that accurately predicts the length value for some weight value in our experiment. We have made some observations and now we want to generalize the observations in to what science calls a law.