Hi need help with the highlighted questions and the graphing exercise on pages 10-13

Laboratory 00 Intro Introduction to Measurements When making quantitative measurements that involve continuous variables, the level of uncertainty must be reported. Better instruments and laboratory procedures will yield results closer to the actual result. It is important to note that obtaining the exact answer is not as important as learning how to report a experimental value along with the level of uncertainty. In other words, you must be honest when reporting values. OBJECTIVES In this activity, you will interpret and analyze data. PART 1 - RULES AND DEFINITIONS 1. Measurement error The difference between the experimental value of a quantity and the accepted value of the quantity. Error 2 Experimental value Accepted value Example 1 The accepted value of 'It to seven decimal places is 3.1415926. If a circumference experiment yields a value of 11: to be 3.16, then what is the error in the measurement? Solution Error = 3.16 3.14 = 0.02 2. Relative error The error of a quantity divided by the accepted value of the quantity. If XK is the actual/accepted value and xE is the experimental value, then IIK 'xEl x Relative error = K _ lxK TIEI 3. Percent error x 100% This is used when comparing an experimental value to xx an accepted value. The percent error provides an idea of the accuracy of the measurement. Example 2 What is the percent error in the measurement of it = 3.16? Solution We found the error to be 0.02 in the previous example. The relative error is then % = 0.00637 to a few decimal places. The percent error is then: 0.00637 x 100% = 0.64% Laboratory 00 Intro 100%. When comparing two experimental values, the 'x, ixli Percent difference = (x1 + x2) 2 percent difference provides a measure of the precision of the experiment. Notice the denominator is the average of the values. Personal errors are mistakes made by the experimenter when taking data or in calculating. Systematic errors result from incorrectly calibrated equipment, poor laboratory habits, andlor incorrect zero point positioning. Repeating the measurement will not reduce the error. Systematic errors cannot be analyzed using statistics. Random errors that are produces by unpredictable and unknown variations. All personal and systematic errors can be eliminated, but some random errors will remain. Random errors can be analyzed using statistics. Accuracy The ability of a measurement to match the actual value of the quantity being measured or how close the measurement is to the true value. Example 3 If the actual value of gravity is accepted to be 9.8 mfsz, then which measured value is more accurate, 9.7 m/s2 or 9.5 m/sz? Solution 9.7 m/s2 is the correct answer since it is closer to the accepted value. 9. Precision The ability of a measurement to be consistently reproduced. The number of signith gures (discussed below) in the reported value indicates the level of precision of the measuring instrument. Small random errors lead to higher precision. Example 4 Which group of measured values has a greater precision, (25 m, 26 m, and 24 m) or (22 m, 28 m, and 32 m)? Solution (25 m, 26 m, and 24 m) is a more precise grouping since the repeated measurement is closer in each case. Accuracy vs. Precision Consider the three images below. Ten shots are red at a target three separate times. Each shot is considered a single measurement. The goal is to hit the target's center. Laboratory 00 Intro Case 1 - This data set is not precise (the repeatability of the measurements is low). None of the measurements are accurate, though the average of the data set may seem accurate (it may land near the center). Arbitrarily chosen measurements should have high percent difference. Without precision, the data set is not reliable. This is an example of using a tool beyond its limit or beyond the abilities of the user, such as firing too far from the target, or trying to measure the thickness of a mosquito's wing with a meter stick. Case 2 - This data set is precise, but not accurate. The repeatability of the measurements is high (they are grouped closer together). The average of the data set is far from the center, though. Arbitrarily chosen data points will have low percent difference, but the average will have a high percent error. This is an example of a systematic error, such as incorrect sighting of the device, or not zeroing the tool properly. Case 3 - This data set is precise and accurate. The measurements are repeatable and the average is near the center. Arbitrarily chosen data points will have low percent difference, and the average will have a low percent error. 10. Significant figures - All the digits in a measurement that are certain plus one that is estimated. Rules for counting significant figures: a. The most significant digit is the leftmost nonzero digit. In other words, zeros at the left are never significant. b. If there is no decimal point explicitly given, the rightmost nonzero digit is the least significant digit. c. If a decimal point is explicitly given, the rightmost digit is the least significant 3Laboratory 00 Intro digit, regardless of whether it is zero or nonzero. d. The number of significant digits is found by counting the places from the most significant to the least significant digit. Example 9 - How many significant figures are in each value? Value Number of Significant Figures 232 3 23200 3 0.230 W 4.0012 5 2004 4 203.20 u 0.000030 2 Note that zeros can cause some confusion when counting significant figures. To clear this confusion, write potentially ambiguous values in scientific notation. Example 6 - How many significant figures does 8000 have? Solution - By the above method 8000 should have one significant figure. Example 7 - How can you report the value 8000 to have two significant figures? Solution - Rewrite 8000 as 8.0 x 103. When measurements are added or subtracted, the answer can contain no more decimal places than the measurement with the left-most decimal place. When measurements are multiplied or divided, the answer can contain no more significant figures than the measurement with the fewest significant figures. Example 8 - 9.001 cm + 2.1 cm = 11.101 cm, but is reported as 1 1.1 cm, since 2.1 ends at the tenths place. Example 9 - 9.001 cm x 2.1 cm = 18.9021 cm2, but is reported as 19 cm2, since 2.1 only has two significant figures.Laboratory 00 Intro 1 1. Precision of the measuring tool - The smallest subdivision that can be read directly. If a single value is measured to be 25.0 cm with a tool of precision 1 mm = 0.1 cm, then the value should be reported as (25.0 + 0.1) cm. Reporting Values and Dealing With Random Errors 12. Mean and Standard Deviation - Random errors have an equal likelihood to be low or high compared to the true value. So, taking the mean x of many measurements X1, X2, ..., Xn is a natural way to reduce the effect of random errors. The mean is defined as and is the best value obtained from all the measurements. (Note: If several values are averaged, a general rule is to assign one more significant figure to the mean value.) Statistical analysis will show that the sample standard deviation 1 ( x , - x ) Vn-14 is a good measure of the precision of the measurements. 13. Standard error a = -" measures the precision of the mean. Vn 14. Reporting the uncertainty - The standard deviation (or standard error if many measurements are made) will substitute as the uncertainty for the mean of many measurements. It is necessary to report it correctly to the reader. Use the following format xto , , or x 1 0 It is important to note that it is necessary to keep no more than one significant figure in the standard deviation and the standard error. (Some texts will say that the standard deviation and standard error should be no more than two significant figures.) Be sure to keep the same decimal place in the mean as in the standard deviation and standard error, even if this means rounding the mean to a lower decimal place (you can remove certainty to ensure the decimal places match). Never add digits to the mean in order to match the decimal place of the standard deviation and standard error (you cannot add certainty). If the standard deviation or standard error is too small, then use the precision of the measuring tool. Example 10 - Given the following measurements find the mean and standard deviation and report it in the correct format. 2.45 m, 2.47 m, 2.43 m, 2.51 m, 2.44 m. 5Laboratory 00 Intro Solution The mean is: 2.45 m+2.47 m+2.43 m+2.51m+2.44 m x = f =2.46 m The standard deviation works out to be 0.0316 In. The correct form for reporting is: 2.46 i 0.03 m since the uncertainty is rounded to one signicant digit (0.03) and the mean is rounded to match the decimal place. In this case, 2.46 ends at the hundredths place, which matches 0.03. If the uncertainty was calculated to be 0.3, for instance, then the correct reporting would be: 2.5 i 0.3 m. Sometimes the standard deviation will be calculated to be too small and will seem to be zero. In this case, we must use the precision of the measuring tool and the measurer's technique to estimate the uncertainty. In other words, the uncertainty would be the smallest value that the measurer can read directly. Propagation of Error It is not entirely trivial how to include the uncertainties in calculations involving more than one quantity with an uncertainty. It is important to use the method of propagation of error. There are two forms of equations that will be discussed here. A) R = )'rx\"y"zC the propagation will be found using the following equation: 2 2 o- 0" 20' 2 Z 2 x 2 2 (Al) R a ?+b y;+c 22 B) R = ax + by + cz the propagation will be found using the following equation: (Bl) of, =a20'f +5-2on +020;2 2 Example 11 The equation for volume of a cylinder is V = m: L , where d = diameter and L = length are the only two measured values. Since the V = volume is to be calculated from measured values, to nd the uncertainty of the volume, you must propagate the error. Laboratory 00 Intro Solution The volume equation matches the form of equation A, since the measured values are being multiplied. The rst step is to put the volume equation in the form of equation A: R = V (calculated value) a = alt/4 (coefficient of the equation) x : d (rst measured value) b : 2 (exponent of the rst measured value) y = L (second measured value) c = 1 (exponent of the second measured value) 2 = 1 (no third measured value) d = 0 (no third measured value) Substituting the above into the matching error equation (A1), we nd: 2 2 2 zzzazn V2 d 2 L2 where the values of d and L would be the mean values. Solving for 0-}, yields the uncertainty of the volume. 9mm Several experiments will require you to construct a graph or a curve. Unless otherwise specied, these are to be done by hand into your notebook. The following items should be considered: 1. The Axes The horizontal axis is known as the axis of the abscissa, and the vertical axis is known as the axis of the ordinate. In most cases, the instructions should illustrate which quantities are to be plotted horizontally and which are to be plotted vertically. Generally, the independent variable, typically horizontal, is taken as the abscissa and the dependent variable, typically vertical, is taken as the ordinate. Often you are asked to plot the ordinate versus the abscissa. For example, if you are asked to plot F vs. x, then you construct a graph with F on the vertical axis and x on the horizontal axis. Always label the axes with the variables and their units. . The Table A table of quantities to be plotted should be made for convenience in plotting and to aid in selecting a scale. The units of each quantity should be identified at the top of the column. . The Scale The scale of the graph is the number of units that correspond to one space or block on the graph paper. The scale should be chosen for both ordinate and abscissa that the curve, when dravm, will extend over most of the paper. Remember that the larger the space in which the data ts, the more precisely the points can be plotted. At the same time, a convenient scale should be chosen which is not awkward. Consult your table to determine a suitable range. Before deciding on a scale, try it out to see if points can be plotted easily. In almost all cases, values should increase from left to right and from below to upward. Indicate the scale plainly by numbering the divisions. You do not have to number every division. Laboratory 00 Intro . The Plotted Points The plotted points should be small but identiable. Ifseveral curves are to be drawn on the same set of axes, use different identication around the points of each curve, circles for the rst, triangles for the second, squares for the third, etc... This should be done before attempting to draw the curve itself. . The Curve After the points have been plotted, a curve corresponding to the theoretical expectations should be drawn. If, for example, a straight line is expected, it should be drawn in such a way that about one half of the points miss the line on the same side. This is the line of best fit. A linear least squared t of the data can be performed. Use a ruler or straight edge to draw a straight line. . The Title Write a title above the graph and caption below it if necessary. Laboratory 00 Intro For questions (1) (8) use the following information. You have measured the length of a table to be 205.0 cm, 205.8 cm, 205.4 cm, 204.6 cm, and 204.9 cm ve independent times. You measured the width of the same table to be 60.1 cm, 60.4 cm, 60.2 cm, 60.0 cm, and 60.5 cm ve independent times. 1) Calculate the mean length L of the table. 2) Calculate the standard deviation of the mean length or of the table. 3) Calculate the mean width W of the table. 4) Calculate the standard deviation of the mean width ow of the table. 5) Calculate the area A = L x W of the table. 6) Using the correct equation for propagation of error, calculate the uncertainty of the area O'A of the table. 7) Calculate the perimeter P = 2L + 2W of the table. 8) Using the correct equation for propagation of error, calculate the uncertainty of the perimeter or of the table. 9) Report the mean length, mean width, area, and perimeter including their uncertainties in the correct format discussed above. Laboratory 00 Intro graphing Exgrg'gg Suppose an experiment were conducted on the stretching of a spring as a function of the force applied to the spring yielding the data in the following table Spring extension Force standard deviation cm cm N (N 0.00 0.06 0.0 0.5 1.25 0.07 1.0 0.5 2.08 1.04 2.0 0.5 3.8? 0.32 3.0 0.5 4.31 0.40 4.0 0.5 5.62 0.79 5.0 0.5 6.42 0.25 6.0 0.5 6.60 0.79 7.0 0.5 8.92 1.24 8.0 0.5 (Note: The values above are held to a few digits without consideration to the rules of reporting values discussed on pages 4 and 5. This is common for data and calculations tables to avoid roundoff errors in future calculations. Following the rules for reporting values, the correct reporting of the value x in the third row of data above would be 3: = 2 :l: 1 cm and the correct reporting for row 4 above would be x = 3.9 i 0.3 cm, and so on.) First, you will need to decide where to draw the axes and what the scale should be. There are no absolute right or wrong choices, but some choices are better than others. Label the axes and mark convenient intervals Next, plot a graph of F vs. x. This means that F is on the vertical axis (ordinate) and x is on the horizontal axis (abscissa). Afx error bars to each point. Draw a straight line of length 20E vertically and one of length 20',r horizontally centered on each point. Next, draw the theoretical curve. The theory states the relationship between F and x is a straight line through the origin. Draw the best straight line you can through the origin and the plotted points. Try to leave about as many points above and below the line. This is the line of best t. (Ask your instructor for the best t ruler). Finally, draw the MAX and MIN lines. Draw two trther lines through the origin with the largest and smallest slopes respectively that are reasonably near the plotted points. These lines should pass through the edges of some of the error bars. The MAX line will have a larger slope and the MIN line will have a smaller slope. 10 Laboratory 00 Intro Find the slopes of the three straight lines on your graph. You may use any two points on the line, but points well separated will provide better precision. To nd the slope, find the \"rise over the run\" the change in vertical over the change in horizontal. Calculate the uncertainty of your line of best t. Statistically, it is accepted to subtract the slope of the MAX line and the slope of the MIN line and divide by 2. This will give the uncertainty of the slope of the line of best t. A plot of the data with error bars is shown below. The slope of the best t is calculated to be 93 N/In. The slopes of the MAX line and MIN line, respectively, are 110 N/In and 82 N/m. The uncertainty of the line of best t is then found by taking the difference of these two slopes and 1 l2 82 2 reporting signicant digits, we write Slope = 90 ilO N/m . dividing by 2. This gives an uncertainty of = 14 N/rn. Considering the rules of 11 Laboratory 00 Intro Force vs. Spring Extension 10 8 . 0 6.0 F(N) 4.0 TT 2.0 0.0 0 2 4 6 8 10 x (cm) How do your hand-generated graph results compare? 12Laboratory 00 Intro Force vs. spring extension 10 Linear Fit for: Force vs. spring extension | Force F = mx+b m (Slope): 0.9452 N/cm b (Y-Intercept): -0.1033 N Correlation: 0.9896 RMSE: 0.4217 N 6 Linear Fit for: F vs x 2 | Force Min F min = mx+b m (Slope): 0.9906 N/cm Force (N) Force Min (N) Force Max (N) b (Y-Intercept): -0.2053 N Linear Fit for: F vs x 3 | Force Max Correlation: 0.9777 F = mx+b RMSE: 0.5956 N m (Slope): 0.8847 N/cm b (Y-Intercept): 0.1221 N Correlation: 0.9863 RMSE: 0.5000 N 2 0- 5 10 Spring extension (cm) 13