Fitting Theory to Data Best Fit: Any viable theory has to be testable. If you can't test the theory by comparing it to actual experimental data, then the theory is not really a theory. The essence of physics is to compare your data with theory. This usually involves tting a theoretical expression to the data representing a smooth curve that shows the values theory predicts. From the example used in the previous paragraph the data suggests that the voltage is proportional to the current data. The equation V = IR is a linear equation of the form, y = mx + b. The variable y is called the dependent variable because it depends on x. The variable x is called the independent variable. When plotting data that relates y to x, we plot the dependent variable on the vertical or yaxis and the independent variable on the horizontal or x-axis. You should follow good graphing procedures as outlined in the previous article on Graphing Data. When we t a linear equation, y = mx + b to the voltage versus current, the variable y = V and the slope of the line m = R the resistance. The intercept in the appendix on Graphing Data should be equal to zero since that is how the problem was setup. In real life the voltage and current measurements rarely fall in a perfectly straight line as plotted. More likely is that the data will look like the data plotted in gure 1 below. For the more realistic data plotted in gure 1, the dependent variable is the voltage and the independent variable is the current. The equation for the straight line joining the data points would then be written as V = mI + b. The differences between the theory and the data are denoted by the deviation d,. The visual best t to the data is the equation that minimizes the deviations of the data from the theory. The plot again indicates that the constant \"b" is zero but this is not always the case but it should be very close to zero. Figure 2 shows three lines that are not best ts curves. ' ,. Curve A passes through three data points but the deviations from the other data points are too large. Similarly, curve C does not minimize the deviations as it passes very near only three of the data points. Although curve B passes through one data point the deviations from the other data points is very large. The approach used in gure 2 is the kind of t that is produced when a student tries to \"t\" the data by \"hand.\" Fitting a function to the data is the best way to nd the variables that give the best t to the data as in gure 1. How do we do a bestt to the data? For this class the best-t is achieved by using the trend line calculated by Excel. The mathematics used is beyond the scope of this class but it is always better than the student doing a manual t of the data using a ruler and doing it by eye. Your turn: Plotting A Graph Do not delete content and the question order. Highlight your answer in yellow color. Lab 1 Graphing Name Read the assignment carefully Graphing and Fitting Data \"A picture is worth a 1000 words.\" A good graph or plot of scientic data maybe worth 10,000 words, but that is only true if the data is plotted correctly with complete specication of the scale on each axis and the units used. Plotting data is useful to reveal the underlying physics. An example graphing of the voltage (V) as a function of current (A) is shown gure 1. There are several things about the plot that every good plot has in common and you should follow when you make plots or graphs of data: 1. The plot is neat and legible. A straight edge (ruler) should be used to draw straight lines. Print neatly so that what is written can be read or use a plotting package such as Excel or Logger Pro if allowed by your instructor. 2. The gure has a title. The gure number is optional but a description of what the plot is about is essential. 3. The axis's have labels that include the units of the data that are plotted. 4. The axis's labels are centered on the axis. 5. Only major tick marks are labeled. (Not every little tick mark as some students want to do) 6. The coordinates of the data points are not written on the plot. People should be able to interpolate the values of each data point without having them written on the plot. 7. The plot of the data lls the plot area. The scale of the plot is expanded so that the plotth data lls the plot area as much as possible. Plots for laboratory exercises in this course should ll the whole page. 8. Unless there is some compelling reason the axis should start at zero. For this course that will always be the case. 5: m..n...~.m_._._ The physics that we learn from this plot is that i. the Voltage is proportional to the current l:_ owing in the circuit. In this case the .. proportionallity constant is the resistance of the " ' ' ';....: circuit or V = IR. If you draw a straight line through the data points, the slope of the line would be the resistance and have the units of volts/amp or Ohm's. Drawing the best straight line is called curve tting. It is the comparison of the data with some mathematical expression that is our theory. It is easiest to t straight lines to data but not all data falls in a straigth line. Adding a trendline will t the data. Your turn: PIorringA Graph Do not delete content and the question order. Highlight your answer in yellow color. 1. Using Excel to make a plot of the Position vs Time given in the following table. The dependent variable is the position (yaxis) and the independent variable is the time (xaxis). This is the measured position of a rock as it falls from the top of a very high cliff and obeys the equations y = Ig. Make sure you label the plot properly. Label the plot and the plot axes with the appropriate labels. 2. Right click on a data point in the graph and add 3 \"Trend line.\" Select a polynomial of order 2. This is a parabolic equation and describes an object falling under constant acceleration. Also check the box that says 'Display equation on the graph.\" Since the acceleration of gravity is 2 times the coefficient of the x' term, what do you nd for the acceleration of gravity. Copy andpaste your graph in the given space below. (50points) Coefficient of the X' Term = ( 5 points) Calculate the acceleration due to gravity (g) using your equation (do not forget the units) ( 15 points) u as, 3. Working with shapes. (30 points) Follow the instructions carefully and the draw the gure in the given space at the end of the instructions. 1. Select the tab \"Insert>Shapes\" in Microsoft ofce word 2. Select a black arrow (with arrow tip) and draw it vertically in the center of this page 2 to 4 inches tall. Don't be tempted to make it fancy. Just a plain black arrow will do. 3. Select a second black arrow (with arrow tip) and draw it horizontally centered on the rst arrow so that it looks like a coordinate system. 4. Select a circle and draw it centered on the coordinate system about 1 inch in diameter. 5. Select a box and draw it centered on the circle with the circle fully contained in it. 6. Add a text box and type \"I did it\" in the text box. Adjust the size so it can be read. You may have to select each object and move to get the desired gure