Explanation how the Pendulum Exhibits Periodic Motion by Ron Kurtus - Succeed in Understanding Physics. Key words: physical science, forces, weight, gravity, bob, periodic motion, frequency, period, pi, clocks, Foucault, School for Champions. Copyright © Restrictions

# Pendulum Exhibits Periodic Motion

by Ron Kurtus (revised 26 August 2011)

A pendulum consists of a weight suspended on a rod, string or wire. When the weight or bob is moved and let go, the pendulum will swing back and forth in a regular periodic motion. The affect of gravity on the bob results in the periodic motion and its length determines the frequency of its swing. Pendulums have been used in clocks for hundreds of years, because the motion is so regular.

Questions you may have include:

• What are some properties of a pendulum?
• What is a demonstration of a pendulum in action?
• What are some applications of a pendulum?

This lesson will answer those questions.

Note: Click the Play button to hear the text being read.

Time = 6 min. 24 sec.

Useful tool: Metric-English Conversion

## Simple pendulum

A simple pendulum consists of a rod or wire attached at a pivot point. On the other end of the rod is a weight or bob. When pulled to the side and let go, the bob will swing down due to the affect of gravity. Inertia will cause it to continue to move back and forth at a definite rate.

### Frequency and period

The frequency of the pendulum is how many times it goes back and forth per second. The period of the pendulum is how long it takes for it to go back and forth one time.

If f denotes the frequency, then the period is T = 1/f.

In other words, if the frequency of a pendulum is 60 cycles per second (or 60 Hertz), then its period is 1/60 seconds.

Likewise, if the period is 3 seconds, then the frequency is f = 1/T = 1/3 cycles per second or 0.33 Hz.

### Interesting properties

A pendulum has some interesting properties, concerning its frequency:

The frequency of the pendulum is dependent on the length of the string or wire. The shorter the wire, the greater the frequency or how fast it goes back and forth.

The frequency is independent of the amplitude of the swing, provided the initial angle is not large. At larger angles, there is a slight change in the frequency.

Also, the frequency is independent of the mass of the bob. In other words a pendulum with a heavy bob will move at the same rate as one with a lighter weight bob. But this only makes sense, since the acceleration of gravity on a falling object is independent of the mass of the object.

## Demonstration

The demonstration below shows a pendulum that starts at an angle of 30°. You can click the Start button to set it in motion.

### Change angle

By stopping the motion and entering another number in the Initial angle box, you can see that the rate of motion or frequency does not change.

Try entering 20.

### Change length

You can also change the relative length of the pendulum to see how it changes the frequency. (The length in the demonstration does not actually change, but the frequency will change, as if the length actually did change according to what is entered in the Relative length box.)

Enter 2 to double the length.

Enter 0.5 to cut the length in half. (You must use decimals. Fractions won't work.)

### Damping

When a pendulum moves, there is some air resistance on the bob and rod or wire. There is also friction at the pivot point. These resistive forces reduce the amplitude of the swing, such that after a while the pendulum will come to a stop. These forces are called damping forces.

In the demonstration above, you may note that the amplitude of the swing gets smaller with time. There is a slight damping factor included in the simulation.

## Applications

The two major applications of pendulums are in telling time and the Foucault Pendulum.

### Clocks

The most common application of the pendulum is to use its regular motion to control the motion of the hands of a clock. This is still seen in the older grandfather clocks. A gear is attached to a weight. Every time the pendulum goes back and forth, it moves a rocker which allow the gear to move one notch. Gears are then used to move the hands of the clock.

Newer versions of pendulum clocks use a spring to move the gear. But the pendulum still controls the rate of motion of the hands.

The length of the pendulum can be adjusted slightly, if the clock is running too fast or slow.

### Foucault Pendulum

Another interesting application is called the Foucault Pendulum. This pendulum will demonstrate the Earth's rotation.

The Foucault Pendulum is a very large pendulum that is often several stories high. The reason it is so large is so that it will keep swinging over a longer period of time. Friction forces often damp a smaller pendulum and cause to finally stop after a relatively short time.

In 1848, Jean Foucault discovered that when a large pendulum swings over a long period of time, the pendulum appears to be changing directions during the day. What is really happening is that the pendulum is moving in the same direction, but the Earth has rotated under the pendulum.

Although there are now Foucault Pendulum's in many locations, the most famous Foucault Pendulum is at the Pantheon in Paris, France. The picture below shows the size of the pendulum and the scale at the bottom to indicate the positions at different times of the day.

Foucault Pendulum in Paris

To explain how the Foucault Pendulum works, consider putting a pendulum exactly at the North Pole or South Pole. While the Earth rotated on its axis, the pendulum would continue to swing in the same direction in space. It would appear as if the pendulum was slowly changing directions, but in reality it is the Earth that is revolving underneath the pendulum.

This same phenomenon will happen at locations other than the poles, except that the reason is not as obvious.

## Summary

A pendulum is a suspended weight that swings back and forth in a regular periodic motion. The length of the pendulum determines its frequency, while the weight of the bob does not affect the frequency. Pendulums have been used in clocks for hundreds of years, because the motion is so regular.

Many things in life swing back and forth, like a pendulum

## Resources and references

Author's Credentials

### Websites

Simple Pendulum - Hyperphysics explanation

How Pendulum Clocks Work - From How Stuff Works

Pendulum Physlet - Java application and equations for a damped pendulum

The Foucault Pendulum - University of Louisville pendulum

Physics Resources

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