Equations
There are many equations to describe a pendulum. One equation tells us that the time period of the pendulum, T, is equal to 2pi times the square-root of L over g, where L is the length of the string, and g is the acceleration due to gravity (which is 9.8 on Earth).
But since all simple harmonic motion is sinusoidal, we also have a sine equation:
Sine equation
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This one says that the displacement in the x-direction is equal to the amplitude of the variation, A (otherwise known as the maximum displacement), multiplied by sine omega-t, where omega is the angular frequency of the variation, and t is the time. In this equation, you start your mathematical stopwatch in the middle - time, t=0, is right in the middle as it swings by.
Finally, you might be wondering: what is angular frequency? Well, angular frequency is the number of radians that are completed each second. A full 360 degrees is 2pi radians, and that represents one full oscillation: from the middle to one side, back to the middle to the other side, and then back to the middle again. You can convert this angular frequency to regular frequency by dividing the angular frequency by 2pi. Regular frequency just tells you the number of complete cycles per second and is measured in hertz.
Example Problem
Okay, let's go through an example. A pendulum that is 4 meters in length completes one full cycle 0.25 times every second. The maximum displacement the pendulum bob reaches is 0.1 meters from the center. What is the time period of the oscillation? And what is the displacement after 0.6 seconds?
First of all, we should write down what we know. L, the length of the pendulum, equals 4 meters. The frequency, f, of the pendulum is 0.25, the amplitude (or maximum displacement), A, is 0.1, and the time, t, is 0.6. Also, g, as always, is 9.8. We're trying to find T and also x.
First of all, to find T, we just plug numbers into this equation and solve. 2pi times the square-root of 4 divided by 9.8. That comes out as 4.01 seconds.
Next, to find x, we first need to calculate the angular frequency. The regular frequency is 0.25 Hz, so just multiply that by 2pi, and we'll get omega. That comes out as an angular frequency of 1.57 radians per second. Now plug numbers into the displacement equation, and making sure your calculator is in radians mode, we get a displacement of 0.08 meters.
And that's it; we have our answer.
Lesson Summary
A pendulum is a weight hung from a stationary point such that it can swing freely back and forth. A simple pendulum is one where the pendulum bob is treated as a point mass, and the string from which it hangs is of negligible mass.
A simple pendulum undergoes simple harmonic motion. Simple harmonic motion is any motion where a restoring force is applied that is proportional to the displacement and in the opposite direction of that displacement. With a pendulum (or any simple harmonic motion for that matter), the velocity is greatest in the middle, but the restoring force (and therefore the acceleration) is greatest at the outer edges.
These equations can be used to solve problems involving pendulums:
Here, x is the displacement of the pendulum, A is the amplitude, omega is the angular frequency, t is the time, g is the acceleration due to gravity (which is always 9.8 on Earth), T is the time period of the pendulum, L is the length of the string, and f is the regular (non-angular) frequency.
Learning Outcomes
Once you've completed this lesson, you should be able to:
- Define pendulum
- Explain the simple harmonic motion of a pendulum
- Identify the equations used to solve pendulum problems
