Amplitude, Period, Frequency



To study oscillations, it is useful to define some terms. Use the pendulum as an example.

1. The maximum distance from rest position is called the amplitude.

2. The swing from one end to the other, and back again, is called a cycle.

3. The time taken to for one cycle is called the period.

4. The number of cycles per unit time (e.g. one second) is called the frequency.

In the above graph, suppose that x is the displacement of a pendulum bob from the rest position. The the amplitude is 1 cm. The time for one cycle is about 2 s, so this would be the period T. The frequency f is then the number of cycles divided by the time taken. Since there is one cycle in 2 s, the frequency is 1/2 cycles per second, 0r 0.5 Hz. The unit for frequency is Hertz, or Hz, which means cycles per second. The period and frequencies are related by

f = 1/T.

Suppose we have two identical pendula, hung side by side. If both are made to swing freely, their period would be equal. Suppose that with my two hands, I pulled both bobs to one side. I let go of one bob first. After half a second. I let go of the other. Both would swing one cycle in the same time. However, they would not swing together in the sense that, when one swings to the right, the other may or may not also swing to the right. If I could measure the displacements of both bobs with time and plot them on the same graph, the graph would look like this

If I let go of both bobs at the same time, then they would swing together. If one goes to the right, the other goes to the right. If one goes to the left, the other goes to the left. They are said to be in phase. If I let go of the bobs at different times, they may not swing together, then their are said to be out of phase. We can use a number to tells us how much the pendula are out of phase. This number is called the phase difference. How do we find this number?

There are a few ways to do it. Look at the peaks in the graph above. The peaks from different pendula are about 0.5 s apart at their closest. One way might be to simply say the the phase different is 0.5 s. It is often useful to think of this as a fraction of the period. This fraction would be 0.5/2 = 0.25. The commonly accepted way is to think of the period as corresponding to 360 degrees, or 2π radians. That is, associate a cycle of oscillation to a revolution round a circle. The phase difference is then defined as the fraction of 360 degrees. In this case, that would be 0.25 x 360 = 90 deg, or π/2 rad.

This association leads to another common way of defining the frequency. It is called the angular frequency. Recall that the frequency f is the number of cycles per second. The angular frequency ω is then the number of radians per second. Since 2π rad is associated with one cycle, the two types of frequencies are related by

ω = 2πf.

Whereas the units of frequency is Hz, the units of angular frequency is rad/s. In the time of one period, there is one cycle, or 2π rad. Since 2π rad is associated with one cycle, the angular frequency is also given by

ω = 2π/T.



Copyright 2010 by Kai Hock. All rights reserved.
Last updated: 18 March 2011.