Question

(3) Re-read the Introduction, giving particular attention to its final paragraph. Now, suppose that physical considerations lead you to expect that measurements on a certain thermodynamic system will lead to the following relationship between the pressure p and the temperature T p %(kT) where k is a constant. Pressure measurements are taken on the system as the temperature is varied, and you would like to perform a curve fit on this data in order to measure the constant k. As you can see, the equation is non-linear so the techniques of linear least squares curve fitting do not immediately apply. Fortunately, however, if you make a correct substitution, you can transform this equation into a linear one and then the techniques you learned in this lab do apply. I. Make a simple mathematical substitution that transforms the equation from quadratic to linear form. Then, show the equation in its new form. Instead of the four averages shown at the bottom of page 10, what four averages would have to be calculated in this case to perform a linear least-squares fit? III. Show how the slope resulting from the curve fit is related to k 18 Physics Laboratory Exercise Data Analysis with Curve Fitting Introduction The analysis of scientific data is often more complex, time consuming, and subtle than designing and running the experiments used to gather such data. Analysis involves organizing the information collected, processing the measurements to produce final results, comparing these results with theoretical predictions, and interpreting them to deduce their meaning and signilicance Experimentation without thorough analysis would generally yield a jumble of unrefined and usele information; thereforc, successful experimental science depends on the analysis and interprctatio of the data, without which definitive conclusions would be difficult or impossible One of the primary techniques of modern data analysis is a mathematical procedure called curve fitting. Research articles in scientific jounals frequently show that curve fitting is a major tool in the analysis. Curve fitting is also used in economics, business, finance, psychology and other ficlds where it is ncccssary to identify underlying mathematical trends within sets of data or correlated In simplest terms, curve fiting is exactly as the name implies: the data is "fit" to some form of mathematical function, where the function may be lincar, quadratic, logarithmic. exponcntial, or something else. Curve fitting also yields specific numerical results (the values of the "fitted parameters") which often have important scientific mcaning In this lab you will study one specific example of curve fitting: linear least-squares fitting Although the fundamental principles you will lcarn can be extended to quadratic fits, exponential fits, etc., we will stick to the linear case both because it is mathematically simple and because it relates directly to the next lab in this series. In addition, non-linear data can often be fit to lincar functions by making appropriate choices for the variables or by using clever substitutions. Basic Probability and Statistics Probability and uncertainty are always important concerns in the analysis of experimental data. This is because all experiments involve some degree of randomness due to "noise recision, or other unpredictable factors. It is of course beneficial to reduce noise and other imp random effects as much as possible, but it is impossible to completely eliminate them. The "'real rld" in which all experiments are conducted is simply too complicated for that Suppose an experimenter makes measurements on the velocity of a falling body as a function of time. Suppose also that data from one experimental trial is graphed as follows

Answer 1

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