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Electronics Symentics


Motion

Pendulum

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?

  • How do your determine its frequency?

  • What are some applications of a pendulum?

This lesson will answer those questions.

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 back and forth due to the affect of gravity.

The drawing below shows the different factors involved in a pendulum.

Simple pendulum

Interesting properties

A pendulum has some interesting properties, concerning its frequency (how many times it goes back and forth per second).

Dependent of length

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

Independent of amplitude

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

Independent of mass

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.

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.

Calculations

Knowing the length of the pendulum, you can determine its frequency. Or, if you want a specific frequency, you can determine the necessary length.

Frequency

The equation to calculate the frequency of a pendulum is:

f = (1/2pi) * sqrt(g/L)

or

f = (1/2π) * √(g/L)

where

  • f = frequency in cycles per second (Hertz or Hz)

  • pi = 3.14 (it is also written as the Greek letter π, but that may not show up in some older browsers)

  • sqrt means the square root of what is included in the parentheses (sqrt is also seen as the symbol √, but that may not show up in some older browsers)

  • g is the acceleration of gravity (9.8 m/s2 or 32 ft/s2)

  • L is the length of the rod or wire in meters or feet

Length

If you wanted to find the length for a given frequency, you can write that equation as:

L = g/(4*pi2*f2)

or

L = g/(4*π2*f2)

For example, the length of a pendulum that would have a frequency of 1Hz (1 cycle per second) is about 0.25 meters.

Problems

Two problems, applying those equations are:

  1. What is the frequency of a 5 kilogram pendulum on a 2 meter wire?

  2. What is the length of a clock pendulum with a frequency of 1Hz in feet?

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. Every time the pendulum goes back and forth, it moves a gear one notch. Gears are then used to move the hands of the clock.

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 is a Foucault Pendulum 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 using damp a smaller pendulum and cause to finally stop after a relatively short time.

Discovery

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

Explanation

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.

In conclusion

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.





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