Fair Test
In order to ensure that a fair test was being carried out I needed to identify what aspects of the experiment needed to be fixed to make it a fair test. The following are variables which need to be kept constant.
- Amplitude of displacement
As long as the amplitude of displacement is 15˚ or under, the period calculated on the basis of the equation I have used above, is accurate to within ½ %. (Roger Muncaster). I found that measuring the amplitude of displacement accurately was not practical and it was very easy for human error to occur when measuring.
I timed how long it took for a simple pendulum of 0.800m in length to complete 30 oscillations when displaced from 5˚ and then I did the same again, but changed the amplitude of displacement to 15˚. I wanted to see how much difference there would be, so I could decide whether it was important to devise some sort of system to ensure the same amplitude of displacement each time. The two times were within 2.00 seconds of each other, at 53.25 seconds and 54.66 seconds.
Therefore I have decided to keep the amplitude of displacement the same throughout the experiment, but it doesn’t have that much effect on the results and so it is not important to ensure that the amplitude of displacement is exactly the same throughout the experiment. I will measure 15˚ to ensure that the amplitude of displacement is small and stays approximately the same throughout the experiment. As long as the amplitude of displacement is always 15˚ or under it will not affect the results to a considerable extent.
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The type of pendulum used in this experiment is a simple pendulum.
The same pendulum will be used throughout the whole experiment, elimination any chance of the pendulum shape, structure or any other of its properties, being a variable in the experiment, therefore making it an unfair test.
- Measuring the length of the pendulum needs to be done in the same way for each length of the pendulum
The length of the pendulum is the independent variable in this experiment. It will be changed and the period for each length of the pendulum calculated. Therefore, the length needs to be measured correctly and so in my preliminary experiment I finalised where the length was to be measured from and to. The middle of the bob to the fixed point is where I planned to measure. A meter rule will be used and the string will be held taut, but not stretched. The meter rule should be positioned against the wooden blocks and the length will be measured along the string from the point it comes out of the wooden blocks, to the centre of the bob. Determining the centre of the bob was what I thought may have presented a problem, but there was a scored line in the middle of the bob and using a vernier caliper I checked to confirm that this line marked the centre of the bob. So the length of the string will always from the fixed point to the centre of the bob as shown in the apparatus diagram.
- Counting the oscillations and using this to calculate period
The period will be calculated by timing how long it takes for the simple pendulum to complete a certain number of oscillations. The time will then be divided by the number of oscillations to give the period, which is how long it takes for the simple pendulum to complete one oscillation. It is important to calculate the period this way, rather than just time one oscillation because as Roger Muncaster explains, “the error which arises through not being able to start and stop the watch when the pendulum is exactly in the intended position is greatly reduced.” The number of oscillations I finally decided to count was not 50, as Roger Muncaster suggests, but 30. This is because 30 oscillations is sufficient to reduce error and also allows me to carry out more readings, because it doesn’t take as much time, which is one of the constraints I face in this investigation.
It also needed to be decided how to count one oscillation. One oscillation is when the bob has gone from one extreme to another and back. I planned on viewing it as it passed the equilibrium position and using that as the start of the oscillation. However, I found that this made it very easy to lose track of the counting and so decided to count one oscillation starting from the left extreme, it passes equilibrium, reaches the right extreme, swings back and as soon as it has reached the end of the 30th oscillation the stop watch would be stopped.
I also decided that it would be necessary to become accustomed to the rhythm of the oscillations before starting to count them. This would be done by counting down to 0, counting 3, 2,1, 0… as the pendulum completed these oscillations and as the bob hit the left extreme on the 0 count, start the stop watch. This would reduce the risk of human error in counting.
Errors and Actions taken to minimise them
There was one particular problem that I came across when trying out my intended method in my preliminary experiment. This would cause an error in the results. It seemed as if the pendulum wasn’t swinging in just one plane, but in circles. This would cause a problem, as the pendulum would not be undergoing the planned oscillation. I found that to correct this problem, minor adjustments needed to be made to the apparatus set-up.
It is necessary to make sure that the wooden blocks have well-defined right angles and are clamped in well. The string of the pendulum should hang at 90˚ when in equilibrium position. This measurement needs to be checked before any readings or results are taken. If not hanging at 90˚, the wooden blocks need to be adjusted, either in the clamp or in relation to one another, so to achieve this. I also realised that it was necessary to make sure that the string is clamped vertically between wooden blocks. I decided that it would be easier to ensure this by making markings on the wooden blocks and then I could check that the string passes through these.
If any problems appeared to occur, such as the one above or if I lost count of the number of oscillations I would discard that result. The pendulum would also be checked before any reading to make sure that it is hanging correctly before every reading. The wooden blocks need to be kept tight, to ensure that the pendulum is only swinging from one fixed point.
Another aspect of the experiment where error may occur is when starting and stopping the stop watch, as reaction time of the person stopping is included in the final time. However, the reaction time is not a significant value which needs to be considered because it is likely that my reaction time will be the same when starting the stop watch to when stopping it. This means that it will not affect the results and so no precautions need to be taken, except to make sure that the stop watch is started when the pendulum begins its first oscillations and then stopped when it has completed its 30th oscillation.
Number and Range of readings to be taken
The number and range of readings to be taken also need to be decided. The length of the string of the pendulum was 1.200m and so I decided that the upper limit for the length of the string would be 1.000m as the string was long enough to achieve this, but any longer than this and it became difficult to hang it off the desk from the clamp without it touching or interfering with anything. I tried to count 30 oscillations when the length was 0.050m but the oscillations became too fast and erratic to keep track of and it was each to lose count of the number of oscillations. So I decided to set the lower limit as 0.100m. I also decided to have 0.100m intervals between the lengths, as this enables me to achieve a large set of results. So I am going to measure the time it takes for a simple pendulum to complete 30 oscillations at 10 different lengths, which will be 0.100m, 0.200m, 0.300m, 0.400m, 0.500m, 0.600m, 0.700m, 0.800m, 0.900m and 1.000m.
I am also going to repeat each length 3 times and then take an average of the time it takes for 30 oscillations. This average will then be used to calculate the period. It is important to conduct repeats so that anomalies can be identified and then not included in the mean taken to plot the graph. Repeats also allow a measure of precision to be made as the more similar the repeats are to each other, the more precise they are.
Safety aspects
There were no safety issues in particular. It is important to make sure that the clamp stand and G –Clamp is secure and to handle them with care as they are heavy and could be dangerous if dropped.
Apparatus list:
Simple pendulum (consists of 1.200m string and a bob)
1 meter rule with 1mm intervals- range 1.000m - error = ± 0.001m
2 x wooden blocks (55mm x 55mm x 15mm)
180º protractor with 1º intervals
Clamp stand
G- clamp
Stop watch (capable of measuring hundredths of second) - error margin ±0.01seconds
Apparatus Diagram:
Method
- Collect apparatus
- Clamp the base of the clamp stand to the bench top using a G Clamp. Ensure the clamp stand is secure and does not move once clamped.
- The wooden blocks should be marked as shown below:
Ensure the line drawn is 90ْ to the edge of the wooden block and is drawn in exactly the same place on the opposite side.
- Clamp the two wooden blocks in the head of the clamp stand, so that there is no gap (or very little) between them. Ensure that the edges of the two blocks meet and the two lines are aligned.
- Loosen the clamp a little, so that the string of the pendulum can be slipped in between the two blocks. The string of the pendulum should be moved along so that it is in line with the lines drawn on the wooden blocks and hangs vertically. Check that the string passes through where the 2 drawn lines on the underside of the wooden blocks too. Adjust the string if needed and when in the correct position, tighten the clamp.
- Now attach the 180ْ protractor (using blue tack) to the wooden blocks. The 90ْ mark on the protractor should be lined up with the vertical hanging pendulum string. Attach so that the protractor is not interfering with the pendulum string, but when standing in parallel to the protractor you can read the angle the string is hanging at to be 90ْ.
- Use the 1 meter rule and placing the end against the bottom edge of the wooden blocks measure and adjust the pendulum (by loosening the clamp) so the length of the pendulum, which is measured from the middle of the bob to bottom of the wooden blocks, is 1 meter. Make the string is held taught but so not stretch at all when measuring.
- Check the length of the string again in the same way and also ensure it hangs 90ْ vertically. If not adjust to provide these conditions.
- Now you are ready to conduct the experiment and obtain the results. Hold the bob and move it 10ْ to the right. The amplitude of 10ْ is measured by looking at the protractor behind the string. Release the pendulum, so it starts swinging. Make sure it is swinging in one plane, so from right to left and not on more than one plane, perhaps in circles. If it is, stop the pendulum and do this again. You will need to make sure that when you measure the 10ْ you are moving the pendulum through only one plane.
- Stand, so that you can see the pendulum swinging from side to side. One oscillation will be taken to be when the bob swings from the left extreme, passes equilibrium to the right extreme and back again to the left extreme.
Count down 3, 2, 1, 0 to familiarise yourself with the rhythm of the oscillations. On the count of 0 start the stop watch. You should count when the bob completes an oscillation.
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Count 30 oscillations. Stop the stop watch as soon as the pendulum completes the 30th oscillation.
- Record the time.
- Adjust the length of the pendulum for the next reading. Remember to measure the length of the string from the middle of the bob to the edge of the wooden blocks where the string comes out. Before moving the pendulum 10ْ ensure the pendulum hangs vertically at 90ْ and the edges of the wooden blocks are at 90ْ. Also measure the length of the pendulum again before starting the swinging of the pendulum to check that the length is correct.
- Carry out the procedure again, as above, changing the lengths from 1m to 0.9m, 0.8m, 0.7m, 0.6m, 0.5m, 0.4m, 0.3m, 0.2m and 0.1m.
- Repeat the experiment for all the lengths 3 times, so that you obtain 3 times for how long it took the pendulum to complete 30 oscillations, for each length.
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Once all the results have been obtained, they need processing. Convert the readings, which have been recorded in minutes and seconds, into seconds by multiplying the minutes (number before the decimal place) by 60 and then adding the seconds recorded. Then take an average for each length, by adding the three times (converted into seconds) and dividing by 3. Take into consideration any anomalies.
- Calculate the period, by dividing the average time for each length by the number of oscillations, 30.
- The period then needs to be squared.
- Plot a graph of period² vs length.
Error Analysis
The equation used in relation to this investigation is:
T² = 4π² L
g
In order to calculate the error that the calculated value of g will have, I need to reassess the error of all the apparatus.
The stopwatch has an error of ± 0.01 seconds, which means that any reading recorded from the stop watch will carry this uncertainty. The meter rule has an error of ±0.001 meters, which also means any length will have an uncertainty of ±0.001meters. The uncertainties of the measurements need to be taken into account to determine the accuracy of the calculated value of g.
I calculated the uncertainty of g, acceleration due to gravity, at the surface of the earth, for each length. This can be seen in the “error analysis” table. The fractional uncertainties are calculated for all the length and the period² associated with each length. These values for uncertainty are added together to give a fractional uncertainty of g. The actual uncertainty of g is then calculated by multiplying actual value of g by the fractional uncertainty of g.
According to my calculations, at no length is the error of g greater than +0.1ms-² which is the upper limit. The lower limit for g is -0.01ms-². Using this data and the value imposed by the gradient of the graph, the upper limit of g is therefore 9.97ms-² and the lower limit 9.86ms-². The difference between the upper limit and lower limit is 0.11ms-². This is 0.11/9.87 x 100 = 1.11%. This suggests that there is not a great uncertainty in the data.
As far as the number of significant figures the final value of g can be determined to, the limitations of the apparatus are not huge, according to my calculations in error analysis and I think 2 decimal places is acceptable considering the accuracy of apparatus used.
Conclusion
The gradient of the graph was found to be 4.00s² m-1. This would represent 4π² / g, therefore I can say that g, acceleration due to gravity at the earth’s surface is 9.87ms-².
The error margin of this value is +0.1ms-², -0.01ms-², according to my calculations.
The gradient was substituted into the equation:
T² = 4π² L
g
The gradient represented the circled part of the equation above and so once the value for the gradient had been found the acceleration due to gravity at the earth’s surface could be determined by 4π²/ gradient.
The value of 9.87ms-² tells us that objects will experience an acceleration of this size at the earth’s surface. By linking the period of a simple pendulum to it’s length and the acceleration due to gravity at the earth’s surface I am able to determine this figure. My calculations assume that g remains constant for all the different lengths of the pendulum, so the period is exclusively dependent on the length of the pendulum. This relationship between the length and period enabled me to determine a calculated value for the acceleration that the pendulum and any other mass for that matter, would experience.
Evaluation
Most significant measurement
Having calculated the error in all the measurements I took, looking at the percentage errors, it appears that the shortest length of 0.100m I measured carried the greatest percentage error of 1.0%. The period was squared in order to produce a straight line, the gradient of which would represent 4π²/g . I expected this value to carry the greatest error as the error was squared; however the error was so small that even after being squared it was less than the error the measurement of length incurred as shown in the table, “error analysis.”
Possible sources of error: systematic and random
There were several possible sources of error.
Reaction time could mistakably be considered as an error, as there will always be some delay in starting and stopping the stop watch. The reaction time could have been measured and then deducted from all the times. However, this will not have had a major influence on the results, as they all included approximately the same reaction time. Another reason why the reaction time is not significant is because there would have been as much delay in starting the stop watch as in stopping it and so the actual time probably would have been much more accurate than anticipated. While planning I decided to calculate the period by counting many oscillations rather than 1, because the error which comes about from not being able to start and stop the watch at exactly the right time is greatly reduced.
Error could also arise from measuring the length of the pendulum. The meter rule enables us to measure to the nearest millimetre and so the actual length of the pendulum could be 1millimeter less or greater than intended. Having calculated the error in each length measured, the shortest length carries the biggest error, so perhaps if a different set of lengths (with the shortest length not being so small) had been used the results would have been more reliable and accurate.
There could also have been parallax error. This is when the object being measured is not on the same level as the graduated surface of the rule. This means that the angle at which the scale is viewed will effect the results. This is a random error as the angle the scale is being viewed at will change from one reading to another. The metre rule may not be calibrated correctly and so will give rise to systematic error. If a steel rule would have been used, as it has much finer markings, it will most likely to promote greater accuracy than a plastic rule.
A simple random error is that which may occur when counting the oscillations. I set 30 as being the number of oscillations to be counted as the experimenter would be less likely to lose count of the oscillations than if there were 50. Having said that, when I was obtaining my results and carried out the experiment there were a couple of occasions when I lost count and had to discard that result and start again. So it is possible that I may have miscounted the number of oscillations at some point, which will affect the results.
The random error is reduced by taking a large number of results as a random error in a small set of results will bear more significance than in large set of results.
Air resistance is taken to be negligible and so is not taken into account.
Discrepancies or anomalies in experimental data
There didn’t appear to be any anomalies in my set of results. I couldn’t see any points that lie far off the best fit line, so much that they require concern.
Variation in repeat readings
I carried out 3 repeats at each length of pendulum. The table shows the calculated standard deviation of the results. The standard deviation is a measure of the variation of the results from the mean. A small standard deviation means reliable results. The closer the standard deviation is to zero the more identical the repeats are, and so the more reliable they are. Looking at the calculated standard deviations, I can see that the standard deviation of the repeats is very small and approximately zero for every length. This small standard deviation verifies that the results were carried out with precision and care and that there are no anomalies, which should be emitted when calculating the average of the repeats. This suggests that the results are reliable and it is acceptable to base a conclusion on them.
Discrepancies between expected results and outcomes
There aren’t any discrepancies between results and the outcome. The best fit line is suitable for determining g as the gradient represent 4π²/g.
Suitability of technique used
The technique was suitable to identify the acceleration due to gravity at the earth’s surface. The small standard deviation between the repeat results suggests precision and the results seem to fit in with the proposed equation, suggesting that the experiment was set up and carried out appropriately to determine the acceleration due to gravity at the earth’s surface. If a greater set of results had been collected or a greater range of lengths of the pendulum had been used then perhaps the final value for the acceleration due to gravity at the earth’s surface, would have been more accurate.
Reliability of conclusion
The graph showing length vs period² shows good positive correlation as the points are all close to the line of best fit. The line of best fit was drawn quite easily, using the centroid as a guide. I was going to work out the product moment correlation coefficient, which would give a figure to how much the points deviated from the line of best fit, but looking at the graph I don’t think this is necessary as they are all so close to the line. The errors were identified and fixed and while carrying out the experiment and collecting data I took care in ensuring everything was precise and accurate by constantly checking apparatus. Having calculated the error and uncertainty in the measurements I can say that the upper limit for g is 9.87ms-² + 0.10 which is 9.97ms-² and the lower limit is 9.87ms-² - 0.01 which is 9.86ms-². There is only 0.11ms-² between the upper limit and lower limit: 0.11/9.87 x 100 = 1.11%. This percentage difference is so small, that I think the conclusion is reliable being based on these results.
Proposals for improvement or further work
One way to better the experiment that I carried out to determine the acceleration at the earth’s surface due to gravity would be to somehow devise a mechanical method of counting the oscillations and also a mechanical way of starting and stopping the stop watch, so to eliminate reaction time of the experimenter having effect on the results. Perhaps a laser beam could be used as a sensor to count the oscillations. If connected to a logger, the number of times the laser beam is interrupted, by oscillating pendulum count be used to determine how many oscillations have been made. The calculated period would be more accurate and precise if error such as the above could be eliminated and the experiment bettered and the margin of uncertainty made much smaller.
The value I have found is a value for the acceleration due to gravity at the earth’s surface only. The earth’s gravitational field varies all over the earth. It would be interesting to see how much it varies and by how much. For example, it is said that at the equator the acceleration due to gravity is different to at the poles.