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11*311111mTmT*3111T1111Z111111111n"|.: Physics-based Extended Project case studies
Pendulum
Initial research
Lily's A-level subjects included physics, maths and further maths. She enjoyed using maths within physics and for her EPQ she wanted to do a project that involved mathematical modelling linked to some collection and analysis of experimental data.
As part of the background reading for her physics A-level, Lily came across several things that intrigued her. She saw a TV programme and read a book about how clocks were used in the 17th century to determine longitude, which made her think about how pendulums have been used in time-keeping and other applications. She read in a physics magazine that g, the acceleration due to gravity, varies over the Earth's surface, which caused her to wonder about the precision of experimental methods for determining g. And one of her textbooks pointed out that the well-known equation for the time-period of a pendulum is actually an approximation, which gave her the idea that such an apparently simple device might be much more interesting than it first seemed.
Lily decided that her project would involve mathematical modelling of a pendulum and she drew on her wider reading as she developed the research question for her project. She wanted to know whether a more precise expression for the time-period of a pendulum, together with an experimental determination of its period using apparatus available in a school laboratory, would be good enough to detect variations in g over the Earth's surface.
For her initial background research, Lily used library and internet resources to find out how pendulums had been used historically for time-keeping and other applications. She extended this research in several directions, finding out how other oscillators are used in time-keeping, and how modern GPS technologies are used to determine position. She found out about different types of pendulum, including some whose complex behaviour inspired recent research into chaotic motion.
Lily also looked into mathematical modelling of pendulums, which made her think that a uniform bar might be a better design for her purposes than a simple heavy bob suspended from a string. Her research question was:
Can a freely oscillating bar pendulum be used to determine the local acceleration due to gravity with enough precision to detect variations in g over the Earth's surface?
Organisation and planning
Lily started preparing for her project in the summer term of Year 12 and the deadline for the final report was April in Year 13.
She needed access to a science lab for her experimental work, so she had to plan her project around the times that labs were not being used for lessons. She decided to work on the design of her apparatus during the autumn term of Year 13 so that she could complete her data collection well before the end of the spring term. She planned to use the time outside the lab to work on her mathematical modelling during the autumn term and to carry out further research into the measurement of g, leaving the spring term for analysing her data and writing up the report.
Mathematical modelling
Lily realised that a mathematical model of a pendulum would need to consider its angular motion, which had not been covered in her maths or physics courses. After using textbooks and websites to study the relevant physics, she was able to put together some equations to describe how the restoring force on a bar pendulum depends on its angular displacement and on its physical parameters.
But then she met a major challenge: using the force equation to obtain an expression for the time period. By talking to her maths teacher and consulting websites Lily discovered that this step required a 'complete elliptic integral of the first kind' in other words, advanced maths that was far beyond her reach at A-level.
Further research on the internet helped Lily to meet this challenge. She found that it was possible to modify the equation slightly and obtain an approximate solution. The resulting equation showed how time period depended on physical parameters such as the pendulum's length, and included terms involving the angle through which it was swinging.
Data collection and analysis
At the start of her lab work, Lily spent several sessions designing her apparatus. She used a set of metal rods of various lengths and cross-sections and experimented with different designs for the pivot. She devised precise methods for measuring a pendulum's length and release angle. And she carried out several pilot tests using different techniques for measuring the time of oscillation.
Lily's apparatus was quite simple, using only items that were already available in her school's physics department. It was easy to put together and to dismantle, so she did not need to leave it set up but could assemble and use it in short sessions when lab space was available.
Lily planned her data collection so as to make best use of the available time. She thought about how she was going to analyse her results, and this helped her to decide which variables to investigate, and how many times each measurement should be repeated. As well as recording data, Lily considered the experimental uncertainties in her measurements.
Having gathered her data, Lily plotted graphs to show how the period of a pendulum depended on factors such as its length and the angle of swing, using statistical methods to determine lines of best fit.
Next she used her mathematical model to generate theoretical graphs showing the pendulums' behaviour, and compared these with her experimental results she wanted to know how good her model was. She researched the use of statistical tests to compare model with theory and discussed whether various tests were valid for the particular situation she was investigating.
Finally Lily used her mathematical model along with her experimental data to determine the acceleration due to gravity at the location of her school.
Conclusion and evaluation
Alongside her experimental work, Lily had carried out further research into gravitational acceleration and its measurement. She gained a good understanding of factors affecting the local value of g, and found some information about techniques for its precise measurement.
Drawing on her literature research and her experimental work, Lily was able to write a well-supported conclusion to her project. She first discussed whether her mathematical model agreed with her experimental results within the limits of uncertainty and then stated her experimental value for g at her school's location. She finished by evaluating her method for determining g, noting some shortcomings of her method and suggesting how they might be addressed, and giving a reasoned answer to her research question.
Supervisor notes
A project involving the collection and analysis of data would be classified as an Investigation/Field Study and entered for Unit 2 of the Edexcel scheme. The other awarding bodies for EPQs (AQA, OCR, WJEC) do not distinguish between different types of project.
For an Investigation/Field Study EPQ, students should go beyond simply plotting graphs of their results. They should look for opportunities to use statistical methods for data analysis, they should consider experimental uncertainties and they should compare their results with a theoretical or empirical model as appropriate.
In any project, it is important to provide a clear, well-reasoned, conclusion. Starting the project with a focused research question as Lily did helps ensure that the work is directed towards finding an answer, leading to a conclusion that draws together the findings of the project in answer to the question.
The project described here used simple apparatus. While EPQ students might benefit from opportunities to use sophisticated apparatus at their school/ college or through contact with another organisation, such apparatus is not a requirement for a successful and rewarding experimental project.
Discussion points
1 How would Lily have applied and extended her knowledge and understanding of physics as part of her project work?
2 EPQ examiners have identified three dimensions in which a project might demonstrate extension:
deepening understanding (e.g. exploring a topic in greater depth than required for A-level);
broadening skills (e.g. practical skills, data analysis, critical thinking);
widening perspectives (e.g. ranging over two or more subject areas; ethical or social implications).
How would Lily's project demonstrate extension in one or more of these dimensions?
3 A strong feature of Lily's project was her use of mathematical and statistical techniques to address a physics research question. Identify the ways in which Lily used such techniques.
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