Another more complex way to consider obtaining ‘g’, will be done using the period of a pendulum swing , of a particular length, modelled by the formula ‘P = 2πv(L/g)’ . ‘P’ would be the time for one complete oscillation of the pendulum, and L would simply be the length of the string suspending the mass, the longer it is, the longer the period of the swing. Having rearranged the formula, I will then calculate ‘g = 4π²L/P²’, which shall hopefully remain roughly constant in my calculations.
Equipment Listing
Experiment 1
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Plasticine Ball - This is fairly dense, reducing the effects of air resistance and hence my systematic error, and will be convenient to use since it will not bounce away upon each drop.
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Measuring Stick - This will simply be used to measure the various dropping heights of the ball, to the appropriate degree of accuracy for this experiment.
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Stopwatch - This will be used to time the duration for each drop, from the moment I let go, until the moment it hit’s the ground, to the nearest hundredth of a second, so it shall be fairly precise.
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Pen - Simply to mark each increment on the wall from which I will drop the ball from.
Experiment 2
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Clamp Stand - To support the string tied on to it, acting as a pivot for it to swing about freely, with minimal friction to reduce any systematic error.
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Weights - To simply hold the clamp stand firmly onto the table, making the structure sturdier, allowing a smoother swing by the pendulum
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String - To be tied around the clamp stand in a simple knot, hanging downwards freely under the influence of gravity. The mass will be suspended at the bottom of it. This string is reasonably strong and so shouldn’t rip under the stress of the weight.
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Plasticine Ball - This will be moulded to the end of the string and shall act as a mass to be swung by the overall pendulum which is created. This is convenient since it will stick to the string firmly without the need for an adhesive, and there will be almost no risk of it falling off the string.
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Scissors - Simply to cut the string to the desired length.
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Ruler - To measure the length of the string fairly precisely, to the nearest millimetre, as this is more than enough precision for this particular experiment.
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Stopwatch - To time the period of the oscillations to the high degree of precision needed for this experiment, to the nearest hundredth of a second. Since the time will be squared in the equation, the slightest errors or deviations from the norm will be magnified considerably, and so alertness when timing will be essential too.
Preliminary Investigation
Before the main investigations begin, I shall first check the reliability and consistency of my equipment in order to determine whether they will provide me with suitable and viable results or not. I will also use this as an opportunity to work out any practical ways to improve the efficiency of my experimentation, as well as to obtain simple ways to reduce my systematic error.
Experiment 1
These results show no major issue with accuracy, as my g value so far is reasonably near to the known value for it. I’m now aware that I should drop the ball from a similar distance to the ground such as what is used above, as the further it is, the lesser my errors.
Experiment 2
The result I obtained is still relatively close to the recommended value for g, however one major source of systematic error has now been identified, which was responsible for creating the inaccuracy in my measurement. I have learnt that the angle from which I swing the pendulum from creates a huge impact on the linearity of the swing, as increasing the vertical amplitude effectively increases the horizontal amplitude of the oscillations, making the pendulum swing unnecessarily side to side, as opposed to just back and forth. As a result to this, the pivot of the string becomes more subjected to friction than normal, and so my results will not reflect the true acceleration of a freely moving body subjected to only gravity’s influence, hence reducing the reliability. I will therefore in my final experiment, aim not to swing the pendulum from angle any higher than roughly 30° from the normal. I will also time the average period for 4 oscillations instead of 20, as having done the experiment I realised that by the time 20 have occurred, the horizontal oscillations have become so great that they become significantly affected by friction, and so a much smaller number of swings deemed to be appropriate.
Final Methods
Experiment 1
1) Find a flat vertical wall and measure 0.2 metre increments ranging from 1.6 metres, to 2.4 metres, using a measuring stick and a pen to mark them.
2) Starting from the lowest increment, drop the Plasticine ball and immediately start and stop the stop watch as it leaves your hand, and when it hit’s the ground. As the heights increase use a chair to stand on to make dropping easier.
3) For each height increment, repeat this 5 times in order to obtain an average estimate for the drop time.
4) Calculate g for each repeat using the formula, and hence calculate an average value for g from all of the repeats from every increment, to obtain a final overall estimate.
Experiment 2
1) Put the clamp stand on the edge of a table, supported by weights to make it sturdy.
2) Make sure the clamps are facing out wards from the table and then cut an arbitrary length of string of reasonable size, and then finally tie it to the clamps tightly with a knot to provide a strong pivot for the swing.
3) Use some Plasticine to make a ball approximately the size of a ping pong ball, and mould it around the end of the string , maintaining the ball shape, with the string now simply going inside its centre. Measure the length from the point where the string is no longer visible on top of the ball, to the point of the pivot at the top of the string using a ruler, and note this down.
4) Start making swings of the pendulum, by dropping the mass from an angle no higher than 30° from the normal, and then timing the period of four back and forth oscillations using the stopwatch, with maximum response time and alertness for pressing start and stop as the smallest deviations make a huge impact on the results.
5) Calculate an average from the 4 oscillations, for just one period. Use this and the length of the string measured (which will always remain constant in this experiment) in the provided formula, and hence calculate g.
6) Repeat steps 4 and 5, ten times in order to use the 10 g values to calculate a final average mean value for g.
Results Tables & Statistical Analysis
Experiment 1
The following table has been made for a graph to demonstrate the linear correlation in my results, with the value for g however remaining constant.
The regression equation generated for the trend line is y = 9.6x, making the gradient roughly equal to the g value. The product moment correlation coefficient for this data is 0.99589, confirming to us the extremely high strength of the correlation, and therefore the consistency of my results.
Experiment 2
Conclusions To Data
To conclude, the above results show to the high degree of accuracy and precision I calculated ‘g’, the second method proving even more effective than the first. My final calculated value for ‘g’ is 9.81, to 2 significant figures. This is virtually the same as its recommended value for the United Kingdom and solely this, due to slight differences in the earth’s imperfect spherical curvature which creates small deviations in the value for g, as its strength varies across different points on earth. One can also clearly see the relationship between distance and ½t², linked by the acceleration constant g in the graphs drawn. The results to my second method are showing very strong consistency due to its small range, incredibly tight standard deviation showing only slight spread in the data, and also my almost negligibly small percentage uncertainty. This experiment was certainly a success, as these results are viable to be used in everyday calculations to an adequate degree of accuracy.
Evaluation Of Experiment
Like mentioned above, overall I believe that this experiment was most certainly a success, which can be affirmed by my statistical analysis to prove so. Furthermore, using two different methods for this experimentation had allowed me to explore different ways to measure this quantity and as a result, I had the opportunity to choose a result which I knew was guaranteed to be more accurate than the other. I believe that I took every possible measure to reduce any systematic error in my results, at every step of the investigation which is another positive aspect. However, there were also definitely times when my results were subjected to inevitable human error. The most significant effect was on the time measured using a stop watch, as my reaction time is only fast to a certain extent, and so the moments which I started and stopped the stopwatch were not exactly the same as the times for it to start and stop moving. Overall I still calculated my values to a reliable degree of accuracy and precision needed, and so only slight negligible deviations from the true and exact answer may have been made
For future experimentation, there are of course ways in which I can improve the reliability, accuracy and precision of my results. I can firstly, some how use a light gate to measure time in my experiments, as this has a much quicker response time than a human, and so will be much more precise. To measure quantities such as length, I could next time use a screw gauge micrometer, giving me much further and more exact increments to measure it from. Finally another possible alternative method of experimentation to obtain g could be done using a ticker timer, which I attempted to use. The idea behind this methodology is that the ticker timer punches indentation in to a moving pressure sensitive strip of paper every tenth of a second. If one suspends a mass at the end of this strip, and allows it to fall freely under the influence of gravity, the ticker timer will respond by making corresponding displacements between each interval of dents, and if once can calculate the change in speed between every point, one can then calculate the acceleration. This acceleration will of course be ‘g’, however in my trial for this experiment I encountered huge errors in my results, which I chose to not include in this paper, due to their immense lack of reliability. The snag of the metal ‘punch’ in the paper, effectively must have slowed it down significantly, and so to not allow the attached mass to accelerate freely and completely under the influence of gravity to it’s full potential. My pendulum experiment however, by far proved to be the most effective of all three methods, and so my final g value was adequate.
