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
After plotting the graphs the correlation between each of the sets of the points was found to be:
These results show that the best correlation of points was between Time Vs √ Length, being R2 = 0.9832.
The equation of the line was also found to be:
y = 0.2087x – 0.0639
.
. .
The relationship between L and T can be calculated to be:
T = 0.2087L – 0.0639
In this investigation, there are many errors that may have occurred.
The most significant would certainly be due to human error. To measure the time of the eight oscillations we used a stop watch. This means that there were at least two errors in each timing period, when the stop watch was started and when it was stopped. This is caused by a person’s reaction time, as it is unlikely that the person would have pressed the button at exactly the time the pendulum was released and when eight oscillations were timed. To try to minimalise this error, we allowed the same person to time each period, allowing the error to be constant.
To try and avoid this, it may be possible to attach the stop watch to the pendulum in some way so the timing is triggered by the pendulum starting to fall.
Another error that needs to be considered is the possibility of the angle not being the same in each timing period. For this investigation to be accurate the angle (see appendices 1) must not exceed 15o. In our investigation, we did not measure the angle, but only approximated it’s proximity to be 15o. If we measured the angle and kept it’s size constant throughout the timing periods, then our results may have been more accurate. To avoid this, the angle should be measured and kept constant for each timing taken.
The length of the string may also have not been exactly the right measurement eg: 20cm. This error would only effect the results slightly, as the error of a few millimetres in somewhat minimal in this investigation. To avoid this, more care can be taken to ensure that the exact measurement of the string is used.
From the graphs, it is also visible that our first readings are slightly obscure. This is most certainly due to the errors already discussed. The most probable cause due to the reaction time of the timer. This may have quickened after the first few readings and become more accurate. To avoid this, practice readings could be taken before the official readings are taken.
The original problem can be extended by investigating whether the weight of the pendulum influences the period of the oscillation. After carrying out some research, I discovered that the period for a simple pendulum does not depend on the mass or the exact angular displacement (as long as it is a small angle approximation), but only depends on the length of the string and the value of the gravitational field strength (gravity).
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.
