There is a very simple way to understand the period of a pendulum.

 

After school, I went to the Y to exercise. I saw many of my students there on the swim team. I took my towel and swung it. Then I folded it and swung it again, noting that the period is smaller. Explained in the locker room without paper or a board how the period of a pendulum depends on length, just using dimensional analysis!

 

In wood shop, students were playing with the power cord hanging from the ceiling. I asked them to estimate the period as it was swinging. Then I held the cord and let it swing again. They noticed that the period was much smaller. A student then let down the cord, making it longer. I pointed out to them the increase in the period. Note that this was a wood shop class, not a physics class!

 

Take a computer mouse and swing it by the cord. Note that the period is longer when the cord is longer.

 

We see that the period T depends upon the length l. How can the period, which is measured in seconds, depend upon the length, which is measured in meters? Well, we note that the period must depend upon the acceleration due to gravity g, for if there were no gravity the pendulum would not swing. The acceleration due to gravity, g, is measured in m/s². Therefore, we must have

 

In order to eliminate the meter, we have to multiply by the length:

We can rewrite this using frequency f, where f = 1/T. We get factors of 2ð. If we rewrite this using the angular frequency ù, where ù = 2ðf, we get

The 2ð comes from the fact that there are 2ð radians in a circle. Using angular frequency instead of frequency gets rid of the annoying factors of 2ð.