# The Pendulum

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Introduction

Maths II Project:

The Pendulum

The following report was prepared for the Pure Maths II topic of Models of Growth. This report aims to investigate the relationship that may connect the period of a pendulum with it’s length.

A pendulum is a body suspended from a fixed point so that it swings back and forth under the influence of gravity. A simple pendulum consists of a weight suspended at the end of a string. The periodic motion of a pendulum is constant, but can be made longer or shorter by increasing or decreasing the length of the string.

To investigate the period of a pendulum, we began by constructing a simple pendulum (see appendices 1). To measure the time for one complete oscillation (the back and forth ‘swinging’ motion), we decided to time eight complete oscillations.

Middle

15.25

15.26

1.91

100

16.07

16.01

16.13

16.07

2.01

* Time taken after 8 oscillations.

We decided to measure eight oscillations and then find an average per oscillation as this is more accurate. If only one oscillation was used, then the person’s reaction time to stop the stop watch would probably be greater than the time of the oscillation.

The average time per oscillation was found by dividing the average of the eight oscillations by eight.

The graph of Time Vs Length can then be plotted to try and establish the rule connecting the period of the pendulum with it’s length.

From this graph there appears to be a logarithmic or power relationship between the period (T secs) and length (L cm). To investigate this, graphs using:

- T Vs Log L
- Log T Vs Log L
- Log T Vs Log L
- T Vs √L
- √T Vs L

can be plotted to determine the relationship connecting L and T.

√length

√Time

Conclusion

The following formula can be derived from these principles;-

At a given place on earth, where g is constant, the formula shows that the oscillation period T depends only on the length, L, of the pendulum. Furthermore, the period remains constant even when the amplitude (the angle) of the oscillation diminishes due to losses in energy such as the resistance of the air,

This property is what makes pendulums good time keepers as they inevitably lose energy due to frictional forces, their amplitude decreases, but the period remains constant.

In conclusion, I found that Time is proportional to √Length and hence found the relationship between L and T to be:

T = 0.2087L – 0.0639

This relationship also supports the formula for calculating the period of a pendulum:

While researching the relationship between the period, length and mass of a pendulum, I found these web sites useful.

http://theory.uwinnipeg.ca/physics/shm/node5.html

http://www.gmi.edu/~drussell/Demos/Pendulum/Pendula.html

http://www.picotech.com/experiments/pendulum/pendulum.html#discussion

http://www.tmeg.com/esp/p_pendulum/pendulum.htm

This student written piece of work is one of many that can be found in our GCSE Forces and Motion section.

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