Pendulum Math

If you did the pendulum experiment described last week, you learned something that may have surprised you. The period (time for one complete swing) only changed if you changed the length of the pendulum. It didn't matter how heavy the weight was, or how far you pulled it back. The first scientist to really study pendulums was Galileo Galilei, an Italian physicist who lived in the 1500's. He was a pretty busy guy because in addition to studying pendulums, he also built the telescopes and discovered the moons of Jupiter! Oh, and also discovered some important laws of physics.

So, the period of the pendulum gets shorter when the length gets shorter, too. Results like this, both the variable (what you change) and the result (what you measure) go up, or go down, are called direct relationships. But this isn't the simplest direct relationship, because when you shortened the pendulum to one-half its original length, the period did not go down by one half, too. In fact, you should have found that it was about two-thirds of the original period.

If you want to learn more about the math behind pendulums, try the experiment described below. You need to know what decimal fractions are, and how to draw a graph, for this to make sense to you. You can write down your results, and make a graph using a computer program like Microsoft Excel. You may need to find an adult to help you with Excel. Now for the experiment: basically repeat what you did in the last pendulum experiment, but use more lengths. In addition to one-half the original length (decimal fraction = 0.50), measure the period for three-fourths (decimal fraction = 0.75), two-thirds (0.67), one-third (0.33), and one-fourth (0.25) the original lengths (1.0). Don't pull the pendulum back too far. Be careful with the measurements of length and period, and write all your results in a table.

When you graph your results, they should look something like the figure on the right, except that you'll have actual numbers on the vertical axis. Now this may not seem very exciting, but it is an important way that scientists learn about the world. They often make measurements, put them on a graph, and look at the results. For example, notice that the line isn't straight - it curves a little. In fact, scientists can analyze data like this and write equations that tell them the relationship between a variable and a measurement. In this case, it turns out that the period is related to the square root of the length. Equations like this are very powerful because they let scientists use math to understand the world around us works.