Statistical Analysis of Data Lab Objective The goal of this "lab" is to learn some of the data analysis techniques that you will be using throughout the rest of the semester and most likely beyond. Specifically, you will learn how to use a data set to create a plot in a spreadsheet program, and you will learn how to draw conclusions about your data from the plot. This lab relies heavily on the content in the "Guide to Data Analysis" handout. Please refer to that handout if you have any questions as you go. Note: In this lab, you will be making two plots in Excel (or a similar spreadsheet program). You will need to attach these plots, along with the columns of data that you entered in the spreadsheet program, and turn them in with the lab. Case 1 Newton's Second Law Introduction We will start with the example from the "Guide to Data Analysis" handout involving a hypothetical experiment to test Newton's second law. Essentially, you will be reproducing the example from the handout. This way you can compare to a known example to check that you are on the right track. As in the handout, we imagine subjecting an object of an unknown mass (let's say, a block) to a single known force and then measuring the acceleration value that results. We then imagine doing this experiment many times for many different force values and recording the data that results. We then organize the data into two columns as shown below. Data Table 1 Force (N) Acceleration (m/s2 ) 1.00 2.22 2.00 3.78 3.00 5.81 4.00 8.43 5.00 10.9 6.00 11.7 7.00 13.9 8.00 16.4 9.00 17.8 10.0 19.2 Questions We know that Newton's second law says that the force, mass and acceleration should be related by the equation = 1. In our experiment, did the force on the block vary or was it constant for all trials? 2. In our experiment, did the acceleration of the block vary or was it constant for all trials? 3. In our experiment, did the mass of the block vary or was it constant for all trials? 4. Given the fact that the force and acceleration varied, while the mass stayed constant, we want to create a plot of force vs. acceleration ( vs. ). Given the form of Newton's 2nd law above, would we expect our plot to make a (i) linear curve, (ii) quadratic curve (parabola) or (iii) square root curve? Recall that the equation of a line is = + 5. What quantity, force, mass or acceleration would correspond to the "y" value in our plot? 6. What quantity, force, mass or acceleration would correspond to the "x" value in our plot? 7. What quantity, force, mass or acceleration would correspond to the "slope (m)" value in our plot? 8. According to the expression for Newton's 2nd law, what should the y-intercept (b) value be in our plot? Plotting For the details of how to carry out the plotting steps, see the Appendix in the "Guide to Data Analysis" handout. 1. Open Excel or a similar spreadsheet program. Enter the exact data table from above (Data Table 1) into your spreadsheet. 2. Create a scatterplot of the data, making sure that the "acceleration" values from the data table are on the x-axis and the "force" values are on the y-axis. 3. Label the y-axis "Force (N)". 4. Label the x-axis "acceleration (m/s2 ). 5. Create an appropriate title for the plot which tells the reader what you are plotting. 6. Use the "trendline" feature to have Excel produce a best-fit line for your data. The equation of the best-fit line as well as the " 2 " value should be present on the plot. Note: Your plot should look very similar to the plot labeled "Force vs Acceleration Plot 3 with Trendline" on the middle of page four in the "Guide to Data Analysis" handout. 7. Attach the plot to this lab report! Analysis 1. Write the equation of your best-fit line below. This equation comes directly from your plot. 2. Given that the mass of the block in our "experiment" corresponds to the slope of the plot, write the best-fit value of the mass of the block below. This is your "experimentally" determined mass for your block. 3. What was the 2 value for your data? Write it below. 4. As was explained in the "Guide to Data Analysis" handout, the 2 value tells you, in general, how strongly correlated the variables that you plotted are. In this case, the 2 value tells you how strongly related force and acceleration are. Our expectation is that force and acceleration are linearly related, as per Newton's 2nd law. Does the 2 value for your data suggest that force and acceleration are indeed linearly related, or does it suggest that force and acceleration are not related? Explain briefly. Case 2 A Model for the Force of Air Resistance You will do one more example of data analysis for a different imaginary experiment. In this "experiment", we imagine wanting to test how the force of air resistance on an object is related to the speed of the object. Background Some of the factors that affect the force of air resistance on an object are (i) the speed of the object, the (ii) surface area of the object, and the (iii) density of the air. One model of air resistance that is often used is a model which suggests that the force of the air resistance is proportional to the speed squared of the object moving through the air, assuming the density of the air and the surface area of the object remain constant. Mathematically, this is written as 2 If this model is correct, then, for instance, if the speed of an object doubles, the force of air resistance on it quadruples. The constant coefficient (often called the "proportionality constant") that relates the force and the speed squared is called the "resistance coefficient", labeled . Thus, our model of air resistance is = The resistance coefficient takes into account the density of the air as well as the surface area of the object. Therefore, the value of will generally be different for each object. However, for a given object in a given room of air, we expect the surface area and air density to be constant, so the value of will be a constant. Imagine doing an experiment where you cause some given object (let's say a parachute of a fixed surface area) to move at different speeds through the same air and then measure the force of air resistance on the parachute for each speed, recording the data in the table below. Data Table 2 Speed (m/s) Force of Air Resistance (N) 1.00 0.41 2.00 3.01 3.00 4.02 4.00 7.03 5.00 10.9 6.00 11.9 7.00 19.8 8.00 22.6 9.00 32.1 10.0 35.8 Questions Recall that our model for air resistance is given by the equation = 1. In our experiment, did the speed of the parachute vary or was it constant for all trials? 2. In our experiment, did the force of air resistance on the parachute vary or was it constant for all trials? 3. In our experiment, did the parachute's surface area and the air density vary or was it constant for all trials? 4. Given your answer to question (3), do we expect that the resistance coefficient varied or was it constant for all trials? 5. Given the fact that the speed and force of air resistance varied, while the resistance coefficient stayed constant, we imagine creating a plot of force of air resistance vs. speed ( vs. ). Given the form of the equation for the air resistance model above, would we expect our plot to make i. A linear curve? ii. A quadratic curve (parabola)? iii. A square root curve? Circle one answer from above and explain briefly. (If you are not sure, plot the data above and see what the curve looks like!) Plotting You should see that since the speed is squared, a plot of force vs. speed would be expected to make a quadratic curve (a parabola). For our purposes, a quadratic curve is not something we want. Remember, we want to be able to find an equation for a best-fit line, so that we may find a slope for our line. However, a quadratic curve is of course not a line and does not have a single slope value. To get around this problem, we will create a plot, not of force vs. speed, but force vs. speed squared! This will "force" the equation to become a linear equation. 1. For each speed in the data table above (data table 2), square the speed and enter the speed squared values into the data table below. Data Table 3 Recall that the equation of a line is = + We will create a plot with the Force of Air Resistance as our "y" value, and the Speed squared 2 as our "x" value. We now expect this to be a linear plot. 2. Based on the equation describing the air resistance model, what quantity, , 2 or would correspond to the "slope (m)" value in our plot? 3. What should the y-intercept (b) value be in our plot? Speed squared (m2 /s2 ) Force of Air Resistance (N) 0.41 3.01 4.02 7.03 10.9 11.9 19.8 22.6 32.1 35.8 4. For the details of how to carry out the plotting steps, see the Appendix in the "Guide to Data Analysis" handout. Open Excel or a similar spreadsheet program. Enter the exact data table from above (Data Table 3) into your spreadsheet. Create a scatterplot of the data, making sure that the "speed squared" values from the data table are on the x-axis and the "force of air resistance" values are on the y-axis. Label the y-axis appropriately. Label the x-axis appropriately. Create an appropriate title for the plot which tells the reader what you are plotting. Use the "trendline" feature to have Excel produce a best-fit line for your data. The equation of the best-fit line as well as the " 2 " value should be present on the plot. Attach the plot to this lab report! Analysis 1. Write the equation of your best-fit line below. This equation comes directly from your plot. 2. Given that the resistance coefficient corresponds to the slope of the plot, write the best-fit value of the air resistance coefficient for the parachute below. (Note that the units of would be N m2/s 2 ). 3. What was the 2 value for your data? Write it below. 4. As was explained in the "Guide to Data Analysis" handout, the 2 value tells you, in general, how strongly correlated the variables that you plotted are. In this case, the 2 value tells you how strongly related force of air resistance and speed squared are. If the air resistance model does indeed describe air resistance usefully, we expect that the force of air resistance and speed squared are linearly related. Does the 2 value for your data suggest that the air resistance model that was given above is a useful model, or that it does not do a good job of modeling the dependence of air resistance on speed? Explain briefly.