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# Period of Oscillation of a Simple Pendulum

Extracts from this document...

Introduction

Andrew Rudhall 11H

Science 1 - Physics

Centre No: 63203                                Candidate No: 7152

Period of Oscillation of a Simple Pendulum

Aim

To find out what factors affect the period of oscillation of a simple pendulum.  I hope to find what these factors are by varying factors such as angle of release, mass of pendulum bob and length of pendulum.

Background Information

• Pendulum

This is a device which consists of an object (pendulum bob) suspended from a fixed point that swings back and forth whilst it is under the influence of gravity.  Due to the constant at which pendulums swing, they are excellent time measuring pieces and eventually led to the invention of the pendulum clock, which was the most accurate time measuring piece at that time.

• Galileo’s Formula

The acceleration due to gravity of an object in freefall was a very important notion for Galileo to realise.  There is a simple law that states the distance travelled in freefall is proportional to the square of the time elapsed.

It was during a Sunday mass in Pisa when he noticed a chandelier above him swinging to and fro.  Using his pulse as a clock, Galileo saw that the period of the swing was independent of how far it swung.  Only the length of the pendulum made any difference to the time required for a swing.

This formula allows the period of a pendulum to be calculated:

P = 2∏√(l/g)

• P is the period of oscillation of the pendulum (in seconds).
• L is the length of the pendulum (in metres).
• G is the acceleration of gravity (on Earth this is counted as 9.8).
•  has the value of 3.14159...etc.

Prediction and Hypothesis

Length is a certain factor, which will determine the time of oscillation of a pendulum.

Middle

This will simply be investigated by placing masses onto the string via a hook and timing ten oscillations as before and then dividing this result by ten.  50 gramme masses will be used, with a maximum mass of 350g.  There will be a total of seven results.  To make the test a fair one, each experiment will have a length of 50cm and an angle of release of 30°.  This should ensure that the results are accurate enough to find out if the mass of the pendulum bob affects the oscillation.  As stated before, I shall draw scatter-graphs etc and draw up tables to see if any patterns emerge.

• Angle of Release

This experiment will be done in a similar style to the others.  Ten oscillations will be timed and as before the answer will be worked out by dividing by ten.  To keep this experiment fair, I shall keep the length at 40cm and the pendulum bob used will be identical each time.  The experiment will be repeated at 5° intervals, ranging from 5° to 100°.  As mentioned before, I shall analyse the results and see if any patterns emerge.  Graphs will be drawn to show any patterns that emerge.

There are no safety precautions that need to be taken into account in this experiment, only common sense should be observed.

Results

In the results section, I shall present information in the form of tables and graphs (e.g. scatter-graphs, bar charts etc).  Hopefully, I shall be able to explain or give reasons for the results that I achieve and why the pendulum acts in this way.  I will also observe any patterns and explain why they are occuring.  I shall also make comparisons to the theoretical answers put forward by Galileo

• Length of pendulum

Here, the results are very good, with no anomalous, unexpected results.  The results that I had were very accurate except for the lengths of 20 cm and 65 cm, which both suffered the worse scores compared to Galileo’s theoretical answer.  They scored a difference of -0.027 and -0.039 respectively.  However, the results were pretty accurate, considering the results were only a few split seconds off being absolutely correct.  In fact one result, 40 cm had a result where the theoretical answer was achieved.  By totalling together the differences between the theoretical answer and my answer, I can see how incorrect the results are as an average.  On average, the results are out by -0.0675 seconds.  This is only a small amount of time, however, if I had my answers exactly the same as the theoretical answers, then the total would equal zero.

I have drawn a scatter-graph to show these results.  The scatter-graph shows the results of my experiment (in red) and the theoretical results (in green).  The green dots at which the red dot cannot be seen, have the best results as this shows that the results are close to the theoretical answer.  Those that are not so close to the corresponding dot, are not such accurate results.

Following this graph is a chart showing the results of the following expression: -

L  t²(Length is inversely proportional to time squared).  This can be rearranged to form a constant figure which will apply to a certain length and time.  It is rearranged to form L = kt².

This can be rearranged so that ‘k’ can be worked out by dividing length by time squared.  The chart gives this as a constant figure that applies to very length or time.

Scatter-graph that shows results of the experiment that investigates length.

Chart that shows how length is directly proportional to time squared.

The results of this chart can be interprted in various ways.  An average for ’k’ is given in red as 0.253.  This applies to every length or time, so therfore make its possible to predict results.

To check that this is correct, I shall apply it to an answer, which I already know.

I shall work out the time for a 60 cm pendulum as follows: -

L  t²

L = kt²

0.6 = 0.253 * t²

0.6 ÷ 0.253 = t²

t² = 2.372 seconds

t = 1.54 seconds

This is the theoretical answer according to my results.  Galileo’s formula gave an answer of 1.555 seconds. I had a time of 1.551 seconds in my results.  As all three answers are very close, I can presume that both the direct proportion method and the Galileo methods is correct.

It would be possible to work out the time of each oscillation for a 50 metre pendulum and would be worked out as follows: -

L  t²

L = kt²

50 =  0.253 * t²

50 ÷ 0.253 = t²

t² = 197.628 seconds

t = 14.058 seconds

Therefore from this formula I would be able to predict the approximate time of each oscillation of a 50 metre pendulum.  By the Galileo method, one would find an answer of 14.192 seconds.

I have drawn a graph on a separate sheet (Figure 1), which displays how length is directly proportional to time squared.  I have drawn a line of best fit; crosses, which are directly on the line, are the most accurate times.  Those crosses, which are furthest away from the line, are the least accurate results.  From this line of best fit, it would be possible to estimate the correct time for a certain length (or visa versa).

• Mass of Pendulum

This table of results shows that my experiments investigating the effect of mass on a pendulum also had good results.  The theoretical answer for this investigation (Where the pendulum is 50 cm long and is released each time at an angle of 30°.  The theoretical answer is 1.419 seconds, whereas I achieved an average of 1.379 seconds.  Each time the results did not fluctuate that much because theoreretically, mass does not affect the oscillation of a pendulum.  I shall now draw a graph, which shows how mass is related to time of oscillation.

It is difficult to plot a graph, which shows how one value stays precisely the same, so this graph shows how the times varied each time the experiment was repeated.  The average comes to 1.379.  Before, it was possible to work out the direct proportion, however, in this experiment; this is inappropriate because for proportion to work, both figures used must change.  From this information, I can ascertain that mass is not connected to time.

• Angle of Release

Conclusion

± 1 second over hundreds of thousands of years.

Furthermore, accuracy would be improved if the experiments were repeated many times until an average was reached whereby it did not fluctuate at all.  Human beings are also not accurate to tell when the experiment starts or finishes.  What would be nedded is a sensor, which could detect when the experiment had started and when it finishes (i.e. its peak height). This could trigger an electronic signal to highly accurate timing equipment.  This however would be very difficult and very expensive to set up.  There would be little point in doing so especially as the theories have been tested many times before and proven to be correct.

Very importantly, though is that the Galileo formula is only theoretical for a vacuum and does not take into account that there is air resistance.  Ideally, it would be best to perform the experiment in an environment where there is no air (a vacuum).  This way, it would be certain that air resistance could have no affect whatsoever on the results, and it could then be called a fair test.  Galileo’s formula has only been tested out on this planet where the gravitational acceleration is 9.8.  This could be tested elsewhere, where there is differing gravitational acceleration.

These are the only ways in which I believe that additional evidence can be found which would support my investigation.  However the measures needed to check them are not necessary.  The only reallistic way in which the investigation can be backed up is by repeating the experiment until the average remains at a constant, and the experiment is repeated whilst in a vacuum with more accurate timing equipment.

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