PHYSC 3622Experiment 2.19 January, 1999
Laboratory Computation
PurposeA purpose of many experiments is to find the relationship between the measured variables. A good way to accomplish this task is to plot the data and then analyze the graph. In some cases, a direct comparison between the measurement and theory can be made. In this experiment, you will learn various graphing techniques and the method of least square fit.
BackgroundGraphing techniques can be used to present clearly and concisely the experimental and numerical results. A variety of commercial graphing and spreadsheet programs have been used in research laboratory. In this experiment, you will use Sigma Plot to perform linear, semi-log, and log-log plots. In many cases, you may want to fit your data to a known theoretical curve and extract relevant parameters from the fitting. If the data points are linearly related, then the process is called linear regression. In general, the data points may not be linearly related and the process of finding a best-fitting curve is called non-linear regression. The technique to be used in determining the best-fitting curve is the method of least squares. In the appendix, you will find more details about the curve fitting. Sigma Plot has a non-linear regression function, which provides you a quick way to fit equations to your data.
ProcedureExercise 1: Interference pattern from a double slit (review Halliday & Resnick, Chapter 41). Using Sigma Plot to graph the intensity pattern of light emitted from a pair of slits. The interference pattern depends on the slit width , the slit separation , and the wavelength of the incident light. Plot the intensity as a function of the scattering angle (from –80° to +80°) for the following values of and .
(1) , (2).
Note that you may need to plot the data points at very fine increments in order to preserve the details.
Exercise 2: Plotting data on a semi-log scale. The data file EXP.DAT contains a measured correlation function g(t) versus the delay time t for small Brownian particles suspended in water at 20oC. Generate a new column, log[g(t)], and plot log[g(t)] versus t with symbols on linear scales for both the vertical and horizontal axes. Plot a simple linear function y=at+b in line and adjust the values of a and b manually such that you can get a ``best-fit” to the data points (use your eye to judge). Vary the values of a and b around their best fitted values and estimate the uncertainties for a and b. Using Sigma Plot to conduct a linear regression and compare the computer fitted results with your manual results. Print the data along with your fits. Theoretically, the measured g(t) is expected to have a simple exponential form g(t) =g0 exp(-2 t). Find the values of g0 and from your fitting results. Plot the measured g(t) versus t and the fitted curve g(t) =g0 exp(-2 t) on linear scales.
Exercise 3: Plotting data on log-log scales. The data file POWER.DAT contains the measured heat transport coefficient Nu (normalized heat flux) as a function of the Rayleigh number Ra in turbulent thermal convection. Generate two new columns, log(Nu) and log(Ra), and plot them with symbols on linear scales. Using Sigma Plot to conduct a linear regression on log(Nu) versus log(Ra), and print the data along with your fit (solid line). In many cases, you can plot Nu versus Ra on log-log scales, instead of plotting log(Nu) versus log(Ra) on linear scales. Using your linear regression results, plot the fitted power law Nu=Ra (solid line) along with the data. Make sure that you have labeled the axes correctly.
Exercise 4: Least square fitting. In many experiments, the data cannot be simply plotted as a straight line whether you use a linear or a logarithmic scale. In this case, one needs to use the method of least squares to perform the curve fitting. Instead of using someone else’s data, you are required to produce your own data in this exercise. The experiment is simple: measuring the extension of a rubber tubing as a function of an applied force. You can find the experimental setup near the power cabinet next to the door of Room 046. A cm scale tape has been fastened to the cabinet for the measurement of the tubing length. Balance weights are used as the applied force. You may increase the weights in 100 gm steps (starting from 0 gm). Be careful that you do not break the rubber tubing. Table the values of the weight W, the tubing length L, and its uncertainty L on your notebook. After you finish the measurements, enter the measured values of W, L, and L into the worksheet of Sigma Plot. Be sure that you are recording the "full" length of the tubing (excluding the length of the weight hanger) and the total weight (including the weight hanger). Generate a new column on the worksheet for the applied force F in Newtons from the measured weights.
An equation of state for an ideal elastic substance such as rubber is given by
F=K [(L/L0) – (L/L0)-2],
where F is the applied force, K is a constant, L0 is the length of the rubber tubing at zero tension (applied force), and is the temperature. The length L0is a function of temperature only. The equation is empirical and is expected to hold in both tension and compression. For our experiment we measure L as a function of F at constant temperature. Plot L versus F on linear scales with the error bars for L turned on. Using Sigma Plot to conduct a non-linear fit to the above equation and print the data along with your fit (solid curve).
QuestionsIn the curve fitting, it is important to justify that the fitted values of the parameters are physically plausible. In Exercise 4, what is the physical meaning of the parameters K and L0? Can you justify your fitting results?
Suppose the functional relation between the dependent variable y and the independent variable t is given by y= [m t2 + b]-1. What kind of graph you should generate in order to produce a straight line? Explain the meanings of m and b in you plot.
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