ERROR AND GRAPHICAL ANALYSIS WORKSHEET

Transcription

1 Student Names: Course: Section: Instructor: ERROR AND GRAPHICAL ANALYSIS WORKSHEET Instructions: For each section of this assignment, first read the relevant section in the Yellow Pages of your Lab Manual. Show only final answers in the tables. 1 Definitions (Appendix D) What is the relative error (uncertainty), expressed as a percent, in the value 3.5 ± 0.2 cm? What is the absolute error (uncertainty) in 12.7 cm ±5%? What is the absolute discrepancy between the value 9.86 ± 0.07 m/s 2 and the value 9.81 ± 0.01 m/s 2? 2 Notation (Appendix D) Identify the number of significant figures in the following values. Next, write each value in scientific notation, and finally, round it to one significant figure. Original Value Number of Sig. Figs Scientific Notation Scientific, Rounded to 1 Sig. Fig Physics 130 Fall 2018

2 Report the following values as a single statement in the form value ± uncertainty, using Appendix D as a guide: Value (cm) Uncertainty (cm) Reported Value, Including Error (cm) The Error in a Single Measurement (Appendix D) The smallest marked division on a measuring tape is 1 mm. If you use this tape to measure the length of a lab bench, what is the smallest possible uncertainty in the measurement? Provide one likely reason why the error should be larger (Note: the reason must make sense considering the actual measurement conditions): A stopwatch displays time to the nearest 1/100 th of a second. What is a reasonable uncertainty for a measurement made with this stopwatch? Briefly explain. 4 The Error in a Repeated Measurement (Appendix D) A student uses a stopwatch, precise to 1/100 th of a second, to make six independent measurements of the time it takes a pendulum to swing through a complete cycle. She obtains the following data: 1.58 s, 1.61 s, 1.56 s, 1.51 s, 1.60 s, and 1.67 s. Fill in the table on the next page, and report your answers to two decimal places. 2

3 Mean (Average) Time Value: (s) Individual Trial Value (s) Deviation from Mean (s) Mean ± Uncertainty: (s) Suppose a second student makes measurements of the same pendulum swing and obtains the values 1.56 s, 1.56 s, 1.56 s, 1.56 s, 1.56 s, and 1.56 s. Give a reasonable uncertainty for her mean value. Briefly explain your reasoning. 5 Comparison of Values Within Error (Appendix D) State whether the following experimental and theoretical values agree within error or not: Experimental Value (kg) Theoretical Value (kg) Absolute Discrepancy (kg) Do they Agree? ± ± ± ± 2% ± ± ± 5 42 ± 10% 3

4 6 Propagation of Errors (Appendix D) For each set of measured values in the table below, compute the final value, along with its error (using the error propagation rules found in Appendix D). Express your answers in the given units, to the correct number of significant figures. Write only your final result in the table. Do not show any rough work in or below the table. Measured Values Compute: Calculated Result with Error: (do not show work) P final = 605 ± 5 P a P initial = 521 ± 5 P a P = P final P initial P = b = 0.45 ± 0.02 cm a = 2b a = d = 11 ± 1 cm t = 0.33 ± 0.05 s v = d t v = T = 0.5 ± 0.1 N m = 1.0 ± 0.1 kg F = T mg F = g = 9.81 ± 0.01 m/s 2 m = 1.0 ± 0.1 kg v = 22 ± 6 m/s KE = 1 2 mv2 KE = 4

5 7 Linearization (Appendix B) The formula given in the first column states how the measured quantities given in the second column are related. In each case, the formula may be linearized to produce a straight-line graph. Complete the table. Note: Assume that the Measured Quantities are the only variables. Formula Measured Quantities x-variable (x) y-variable (y) Slope (m) Intercept (b) F = ma F, m m F a 0 F = k(x x 0 ) F, x x F F = k(x x 0 ) F, x F x T 2 = 4π2 k m T, m T 2 T 2 = 4π2 k m T, m T y = 1 2 at2 + 6 t, y y 5

6 Read the section on the Construction of Graphs on pages C-3 to C-7 in Appendix C. 8 Best-Fit, Maximum, and Minimum Lines (Appendix E) Draw reasonable maximum and minimum lines on the graph below. Label the max and min lines, and label with a (?) any anomalous points. On the graph below, the best-fit line has already been drawn. Add appropriate maximum and minimum lines. Label your maximum and minimum lines, and denote any anomalous points by labeling with a question mark. best fit 6 6

7 9 Calculating Slope and Intercept with Errors (Appendix E) For the graph found on the back of this page, complete the following: 1. Calculate or determine the minimum and maximum y-intercepts and slopes (show work on the graph): b max = max intercept: m max = max slope: b best = best intercept: cm/s m best = best slope: cm/s 2 b min = min intercept: m min = min slope: 2. Calculate the error in the intercept and slope: E 1 = b max b best : E 3 = m max m best : E 2 = b best b min : E 4 = m best m min : δb = larger of E 1 and E 2 : δm = larger of E 3 and E 4 : 3. State the final numerical answers for b and m with uncertainty and units, and to the correct sig. digs: b ± δb m ± δm 4. Now suppose the theoretical equation being graphed is d t = 1 2 at + v o. Calculate the experimental value for the acceleration a ± δa from your graphical slope value. 5. Determine an experimental value for the initial velocity v o ± δv o from your graphical y-intercept value: 6. Finally, check whether your value for the acceleration agrees with the theoretical value of 0.55 cm/s 2 : 7

8 Graph 1: Distance Divided By Time as a Function of Time for a Cart on an Inclined Plane y = x Elapsed Time (s) 8 Distance/Time (cm/s)