# Differences

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+ | ~~NOTOC~~ | ||

+ | ===== Analyzing Experimental Data ===== | ||

+ | |||

+ | Last week we looked into how uncertainty arises when taking | ||

+ | measurements. This week we will investigate how uncertainty arises | ||

+ | within a set of data and how to analyze your results using tools such as | ||

+ | plots and lines of best fit. | ||

+ | |||

+ | Tools such as Excel or Google Sheets are a great asset to scientists to | ||

+ | help make calculations using an entire data set and these tools also | ||

+ | provide easy access to plotting. In this course, using a spreadsheet can | ||

+ | help to decrease the number of calculations that your group needs to do | ||

+ | for each data point. You can use them to calculate the mean across | ||

+ | multiple columns or to use the cell as a variable in any equation you | ||

+ | define. Plotting is another tool that can save your group from multiple | ||

+ | calculations, and it uses all of the data points in your set to create | ||

+ | the result. With enough data points, using a plot to find your result | ||

+ | can yield more precise conclusions than any one measurement. | ||

+ | |||

+ | In this workshop, you are tasked with obtaining the spring constant of | ||

+ | the spring you are given using various masses as your data points. You | ||

+ | will need to use your plot as a tool to obtain this. This will require | ||

+ | you to discuss which equations will be helpful when you plot. You and | ||

+ | your group will discuss the value of considering many data points within | ||

+ | a measurement and why we look at a set of data to draw conclusions. This | ||

+ | workshop emphasizes uncertainty analysis, again, as understanding | ||

+ | uncertainty is a major learning goal of this course. However, this time | ||

+ | your group will be considering a data set rather than a single | ||

+ | measurement. It also adds to that understanding by further exploring the | ||

+ | concept of uncertainty as it relates to a known model. | ||

+ | |||

+ | ==== Research Concepts ==== | ||

+ | |||

+ | In order to be productive in class, it would be helpful to research | ||

+ | before class: | ||

+ | * Force diagrams | ||

+ | * Lines of best fit for linear data sets and the generic equation for these lines | ||

+ | * Ways of representing this kind of data in graphs | ||

+ | * How to construct a scatter plot in Excel or Google Sheets ([[http://sheets.google.com]]) | ||

+ | * How to add error bars to plots in Excel or Google Sheets | ||

+ | * The difference between precision and accuracy | ||

+ | |||

+ | ;#; | ||

+ | **Potentially Useful Equations**\\ | ||

+ | $F_{spring}=kx$ and $PE_{spring}=\frac{1}{2}kx^2$\\ | ||

+ | where k is the spring constant and x is the displacement from the spring's resting position | ||

+ | ;#; | ||

+ | |||

+ | ==== Data Analysis ==== | ||

+ | |||

+ | **Part 1 -- Discussion of Model** | ||

+ | |||

+ | Your goal will be to find the spring constant of the spring that your | ||

+ | group is given using a ruler and set of masses. But first, you should | ||

+ | discuss how you will accomplish this by using the equipment and the | ||

+ | physics knowledge that you have. To help guide your discussions, | ||

+ | consider: | ||

+ | * What is the force diagram for this system? | ||

+ | * Using those equations, what variables do you already know? | ||

+ | * What variables can you measure with the given equipment? | ||

+ | * What variables do you need to find with equations? | ||

+ | |||

+ | These questions are important to consider before deciding your | ||

+ | procedure. These can help your group create a plan for your experiment, | ||

+ | since the overall goal of the experiment (finding the spring constant in | ||

+ | this case) may not be possible by direct measurement. | ||

+ | |||

+ | **Part 2 -- Take Your Measurements** | ||

+ | |||

+ | Now that your group has discussed what measurements to make, you should | ||

+ | conduct your experiment. Record your data in a spreadsheet. Don't worry | ||

+ | about any calculations you need to make just yet. Focus on recording the | ||

+ | data that you can measure directly first. Some questions to consider | ||

+ | before taking your measurements: | ||

+ | * What is the precision of your measuring tools? | ||

+ | * How did you choose to measure the displacement? | ||

+ | * Is there another way in which you could measure the displacement that would make your results have less uncertainty? | ||

+ | |||

+ | **Part 3 -- Calculations and Plotting** | ||

+ | |||

+ | Plot your data as a scatter plot. Talk with your group about which | ||

+ | variable should be on the x-axis and which variable should be on the | ||

+ | y-axis. If your results are linear, add a line of best fit to your | ||

+ | results. What is the equation for your line of best fit? | ||

+ | |||

+ | Recall the equation you found using your force diagram. Can you | ||

+ | rearrange this equation to look like your line of best fit? If so, | ||

+ | consider the questions below: | ||

+ | * What does your slope represent? | ||

+ | * What does your y-intercept represent? | ||

+ | * Do these results make sense? | ||

+ | * Can you find the spring constant from your plot? | ||

+ | |||

+ | **Part 4 -- Representing Uncertainty in Plots** | ||

+ | |||

+ | We have discussed uncertainty in Workshop 1, but how does the | ||

+ | uncertainty in our measurements translate to a plot? Error bars are the | ||

+ | graphical representation of the uncertainty value that we assign to | ||

+ | measurements. | ||

+ | |||

+ | Discuss with your group the uncertainty on your x-axis and y-axis | ||

+ | variables. Then, add error bars to your plot. | ||

+ | |||

+ | **Part 5 -- Taking More Data** | ||

+ | |||

+ | As we've discussed, the precision of your measurement often depends on | ||

+ | the measurement device and how the measurement was taken. In some cases, | ||

+ | the measurement you take will be the same each time it is taken, while | ||

+ | others will show some variation between data points. If you find that | ||

+ | the variations between points are larger than the precision of your | ||

+ | instrument, you can increase the precision of your data by taking | ||

+ | multiple measurements. | ||

+ | |||

+ | Repeat the measurement done in //Part 2//, two more times**.** Now you | ||

+ | will have a total of three datasets. Plot a line for each new dataset on | ||

+ | the same graph as before (so that you will have three lines on one | ||

+ | plot). Is there a trend between your lines? | ||

+ | |||

+ | Calculate the mean for each data point. On a new plot, create a line | ||

+ | that uses the mean values from all three of your measurements (remember | ||

+ | to use your tools in Excel/Sheets for this!). Use this to find your | ||

+ | final conclusion for the spring constant. Your group should discuss what | ||

+ | the equation for the line of best fit means and if it makes sense. Be | ||

+ | sure to also consider if the size of your error bars should change now | ||

+ | that you are using the mean value instead of a single measurement. | ||

+ | |||

+ | At the end of the day, you will be turning in your notebooks. Tutors | ||

+ | will put particular emphasis this week on your figures, your discussion | ||

+ | of the plan for your measurements, and the discussion of the | ||

+ | equation/model to find your results. | ||