Lab 1: Error Analysis and Data Studio

Transcription

1 Physics Sonoma State University Lab 1: Error Analysis and Data Studio Part I: Error Analysis Errors Knowledge of the physical world is obtained by experimentation which involves people. As introductory experimenters you expect errors, but even advanced scientists know that you cannot determine experimental quantities with zero uncertainty. Techniques will be given for you to express your experimental results, the uncertainty of your result, and ways to compare numerically to known values. In addition you might need to calculate a result from two or more experimental quantities. Each will have its own uncertainty, and the uncertainty of the result is determined by the individual uncertainties. These techniques are well known and supported by theory. In each experiment you will have a result with its associated uncertainty. What sorts of errors can be introduced and why? 1. Systematic Errors Systematic errors are errors associated with the particular instruments or techniques used to carry out the measurements. Suppose you have a digital scale that has not been zeroed, but has a 1 gram offset. All mass measurements then will be too high by one gram. This is a systematic error. If you notice, it would be easy to correct for. With complex instrumentation it might be difficult to determine if it was malfunctioning, or incorrectly calibrated. A pressure gauge might be reading incorrectly. It may not be obvious until you compare your results to known values. Systematic errors will shift all your measurements to one side. All will be either too high or too low. They can be hard to correct for unless you can compare the measurements to known calibrated instruments. Experience will help you become aware of them. 2. Random Errors These are errors resulting from uncontrollable fluctuations in the experimenter s skill or in the equipment operation, from the finite precision of instruments, and from slight variations in the quantity being measured. Random errors can be dealt with using well-defined statistical methods. Random errors don t influence your results in one direction, like systematic errors. Sometimes your measurement is slightly higher than true, sometimes slightly lower. 3. Mistakes Illegitimate errors are blunders or mistakes. They are easy to determine and should not be addressed in your reports, but corrected for.

2 Accuracy and Precision If your result is known to a high degree of certainty then you have a precise result. If your result is very close to a known value then you have an accurate result (small % difference). Significant Figures The significant figures of a measured or calculated quantity are the meaningful digits in it. When you record data keep in mind significant figures. The main violations occur with calculator computations. You will often be presented with a ridiculous number of decimal places from a calculator. Before you write them all down think about the physical measurement and calculations, i.e., how many are significant. When you are given a number, any digit that is not zero is significant. Thus 549 has three significant figures and has four significant figures. Zeros between nonzero digits are significant. Thus 4023 has four significant figures. Zeros to the left of the first non zero digit are not significant. Thus has only two significant figures. This is more easily seen if it is written using scientific notation as 3.4x10-5. For numbers with decimal points, zeros to the right of a nonzero digit are significant. Thus 2.00 has three significant figures and has two significant figures. For this reason it is important to keep the trailing zeros to indicate the actual number of significant figures. For numbers without decimal points, trailing zeros may or may not be significant. Thus, 400 indicates only one significant figure. To indicate that the trailing zeros are significant a decimal point must be added. For example, 400. has three significant figures, and 400 has one significant figure. Exact numbers have an infinite number of significant digits. For example, if there are two oranges on a table, then the number of oranges is Defined numbers are also like this. For example, the number of centimeters per inch (2.54) has an infinite number of significant digits, as does the speed of light ( m/s). When numbers are added or subtracted, the number of decimal places in the result should equal the smallest number of decimal places of any term in the sum or subtraction. When multiplying several quantities, the number of significant figures in the final answer is the same as the number of significant figures in the quantity having the smallest number of significant figures. The same rule applies to division. Statistical Quantities We try to make a number of similar measurements x i (i=1 to ) of the same quantity. The best guess of the true value is the mean. The scatter in the measurement is given by the standard deviation, and the uncertainty is given as the standard error (as called in EXCEL) or the standard deviation of the means. Mean of trials = i=1 1 2 Standard deviation σ = ( xi x) x i i= 1 2

3 Standard error (SE) or experimental uncertainty = σ σ Fractional standard error or standard deviation =, x such that x = ± SE exp result known quantity % difference = x100, where known quantity is often the value that the known quantity theory predicts. Expressing Your Result * There are also specific rules for how to consistently express the uncertainty associated with a number. In general, the last significant figure in any result should be of the same order of magnitude (i.e., in the same decimal position) as the uncertainty. Also, the uncertainty should be rounded to one or two significant figures. Always work out the uncertainty after finding the number of significant figures for the actual measurement, for example, 9.82 ± 0.02, 10.0 ± 0.5. The following numbers are all incorrect ± is wrong but 9.82 ± 0.02 is fine ± 2 is wrong but 10.0 ± 2.0 is fine. 4 ± 0.5 is wrong but 4.0 ± 0.5 is fine. In practice, when doing mathematical calculations, it is a good idea to keep one more digit than is significant to reduce rounding errors. But in the end, the answer must be expressed with only the proper number of significant figures. After addition or subtraction, the result is significant only to the place determined by the largest last significant place in the original numbers. For example, = should be rounded to get 90.4 (the tenths place is the last significant place in 1.1). After multiplication or division, the number of significant figures in the result is determined by the original number with the smallest number of significant figures. For example, (2.80) (4.5039) = should be rounded off to 12.6 (2.80 has three significant figures). *To determine the correct number of significant figures in our results we will almost always use statistical methods. First determine the uncertainty, round that to one significant figure, then round the result (normally the mean of a number of measurements) to the same decimal place. This is reasonable unless the leading digit is a one and the second digit is a 5 (for instance, 0.15). To drop the 5 or round up to 2 would be a large percentage change. 3

4 PART II: Data Studio Through the semester we often use the Data Studio (PASCO 750 ScienceWorkshop) data acquisition system to collect data. Data Studio software controls the interaction with the sensors and not only can acquire data but also has the option of entering independent data into a table and graphing it. A secondary program, such as Excel, can be used for further analysis. In this part you will use the software to set up and acquire data in a simple situation as an introduction. Today you will explore data studio and the motion sensor capability. We will be examining velocity and acceleration. The motion sensor emits pulses of ultrasound that are reflected from the moving cart back to the sensor. The sensor measures the time it takes for the pulses to move out and reflect back. By knowing the speed of sound in air (default is 344 m/s) the distance is calculated. The average velocity can then be calculated between points, and from these the average acceleration. 1. Turn on the interface box. The switch is on the back. 2. Locate the physical sensor at your desk. 3. Plug in the sensor. If it is a digital sensor (round plug) it will plug into one of the left ports, an analog sensor will plug in on the right side. Use Channel 1 port unless there is a problem with it. The motion sensor has two connectors. Plug the yellow one into CH1, the other one into CH2. 4. Double click on the Data Studio icon on the desktop. ow you need to connect the program. 1. Choose Create Experiment and a picture of the interface will appear. Because you would like to use a Digital Sensor, click the first port, CH1. 2. A list of all the digital sensors appears. 3. Double click on the Motion Sensor. The Set-up window opens with several options for measurement and sampling conditions selected. Leave it as is. Opening Graphs 1. On the left side of the window you will see a list of the types of data that will be acquired (position, velocity, and acceleration) and a list of possible displays. Drag the Position icon from the Data window and drop it on the Graph icon in the Display window. A graph window for position should then open up and a new label, Graph 1, should appear under the Graph icon. 2. Drag the Velocity icon and drop it onto the Graph 1 icon to place a second graph in the same window. In each case time is on the x-axis. (You could choose to display acceleration if desired.) ow you are ready to take data. The sensor is connected, and the software is set up and configured. Data Collection Data collection starts by pressing the START button on the toolbar. Data collection can be stopped by pressing the STOP button (The START button changes into a STOP button). Data 4

5 collection can also be set to start or stop after a certain amount of time, or a critical value is reached. (This can be chosen in the Set Up window under Sampling Options.) Each data set is listed as a Run # under the data window. Individual data runs can be graphed, and data runs can also be deleted by selecting and pressing DELETE under the Experiment menu. Practice collecting data, viewing and graphing it, and deleting it, while the track is level. If no graphs are set up drag the data from a single run onto a Graph icon in the Displays window, to graph a single run of position, and velocity. Delete single runs. (Experiment Menu > Delete last run; OR in Data Window, highlight run (click once) and press delete key). Delete all runs (Experiment Menu>Delete all data runs). Delete a graph and notice that the data run has not been deleted. The data can be easily graphed again. Sensor Configuration The parameters for collecting measurements will sometimes need to be changed especially the data collection rates (Sample Rate). Each of the sensors has a default rate for which it will collect data. The motion sensor, for example, collects data at a sample rate of 10 Hz, or 10 points/sec. For many applications, this is fine, but for some, you want a higher rate of collection. For a single motion sensor, it can collect data as slowly as 5 Hz and up to 120 Hz. (ote that the higher sample rates limit the distance at which the sensor can detect motion.) With two motion sensors, maximum sample rate will be 25 Hz for each sensor. Changing configuration Display the Experiment Setup window by clicking the Setup button under the toolbar. Change the sample rate if necessary. Measurement options are in the tab labeled Measurement. Choose which of the following measurements to display: position, velocity, and acceleration. Any of these measurements can always be displayed or removed after a data run is taken, since the velocity and acceleration for the motion sensor are determined strictly from the position information. They will disappear or reappear in the Data window. Data Analysis with Graphs Become familiar with the actions of the following buttons on the graphing toolbar below. szasdfommasdfom Each of the buttons on the graphing toolbar can be identified by holding the mouse over the button until the hint appears. Use the Scale to Fit button ( szas ) to fit full graph data into window. You will probably use this button every time you use Data Studio. 5

6 Use the Zoom Select button ( sdfommasdfom ) to zoom into a particular section of the graph. Use the Smart Tool to determine values of particular data points. Drag the square onto the data curve, and values will be displayed in color instead of black and white. Click and drag corner of square to display changes in x/y values from previously selected point. Use the Fit button to create a Linear Curve Fit ( ) to the data. ote that the curve fit is adjusted to the entire data set. To apply the curve fit to a region of the data curve, select just that region by clicking and dragging diagonally until the region of interest is selected. Important: When a particular data run is chosen on the graph, it will be highlighted in yellow, and a curve fit associated with it will apply to the entire data curve. After selecting a section of the data to apply the curve fit, it will revert to the full data set if any place on the graph is clicked. To keep a partial curve fit, select the subset of data on the graph, then click on a different data run in the graph legend. The subset will remain highlighted yellow, and the curve fit will follow that portion of the curve. Use the Show Selected Statistics button ( ) to show statistics for a data curve or a set of points within a data curve. The drop down menu adjacent to it selects specific statistics. Selecting a subset of data for statistics works similarly to the way the Linear Curve Fit works above. Delete the curve fit by selecting Linear Fit run in the Data window, and deselecting Linear. Use the Slope button to determine the slope at any point on the graph. ote that this only gives the slope of the tangent to the graph at any particular point, and thus differs from the slope given by a curve fit. Move objects such as the legend and curve fit data windows around the graph screen by clicking dragging them. Manual Data Entry Data can also be entered into a table manually. Choose Experiment Menu>ew Empty Data Table. Enter the data in the columns, and click and drag the table from the Data window to a Graph to create a graph. Change column headings by double clicking on the Editable Data icon in the Data window. Saving Work ormally you do not need to save your work. For peace of mind you can save it periodically. Be sure to determine what folder it is saved to, or if it is saved to the desktop. In the event that the program crashes, you can recover to your previously saved point, without redoing all of the experiment. Go to FILE>Save Activity As to save the activity. 6