5. Is it OK to change the spacing of the tick marks on your axes as you go across the page? a. Yes b. No - that screws up the analysis of the data.
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1 Name: Significant Digits, Unit Conversions, Graphing and Uncertainties in Measurements =========================================================== Choose the best answer. (30 pts total) Significant Digits, and Converting Units 1. A student did the calculation, = x, in their calculator and read the result x = , from the calculator screen. How should the student record the result, x, to have the correct number of significant digits? a b. 113 c d. 110 e Express 3300 g/cm 3 in kg/m 3. (Note that the trailing zeros in 3300 are merely place holders and are not significant digits. For the zeros to be significant, the number would have to be written as 3.300x10 3.) a. 3.3 x 10 2 kg/m 3 b. 3.3 x 10 9 kg/m 3 c. 3.3 x 10 3 kg/m 3 d. 3.3 x 10 6 kg/m 3 3. Why do we keep 1-2 extra (non-significant) digits in the middle steps of our calculations and only round to the correct number of significant digits in the final step of our calculation? a. to reduce rounding error b. because we like hitting lots of buttons on our calculator c. We don t keep extra (non-significant) digits (or units) because we like to have errors in our results. Graphing 4. When drawing a graph, it is best to draw it small, so it takes only a small part of the graph paper and the data points are crammed together. In physics, we like to maximize the error in our graphs and measurements and make things hard to read. a. True b. False 5. Is it OK to change the spacing of the tick marks on your axes as you go across the page? a. Yes b. No - that screws up the analysis of the data. 6. Which of the following should you include in the axes labels of your graphs? a. units d. Answers a & b b. symbol/variable for what is being plotted e. Answers a, b & c c. word description of what is being plotted 7. Do not mix units in a graph or they won t cancel properly in the slope (Δy/Δx). Use the same type of units for the same type of number on both axes. Which of the following sets of units is NOT ok to use? [Remember: N = kg m/s 2 ] a. y-axis = N and x-axis = m/s 2 b. y-axis = g cm/s 2 and x-axis = cm/s 2 c. y-axis = N and and x-axis = cm/s 2
2 8. Titles for your graphs a. are not always necessary. b. can be written with symbols/variables only. c. are a word description that describes what is in your graph and are always required. 9. Assuming all data points are of equal quality, which of the following is the best fit line? a. b. c. d. 10. When it is acceptable to use a data point as a slope point? a. if the data point appears to fall upon the line b. if the data point appears to fall upon the line and if it falls at the extreme end of the page c. NEVER, because this tends to influence the positioning of the best fit line. 11. To find the best possible value for the slope of a graph (while minimizing uncertainty), the two slope points should NOT be data points, lie at the intersection ( corner ) of the gridlines, and, as much as reasonably possible, lie at the extreme ends of the paper. If you can t have it all, which is the first thing you give up? a. slope points at extreme ends of the page of paper b. slope points at the intersection ( corner ) of the grid lines c. ensuring that the slope point is not a data point
3 12. Which is the incorrect formula for calculating the slope (Δy/Δx) with the slope points (x 1, y 1 ) and (x 2, y 2 )? a. (y 1 y 2 ) / (x 1 x 2 ) b. (y 2 y 1 ) / (x 2 x 1 ) c. (y 1 y 2 ) / (x 2 x 1 ) 13. How do you find the y-intercept of your graph if your graph does not include the origin (0,0)? a. See where your best fit line crosses the vertical axis. b. See where your best fit line crosses the horizontal axis. c. Use the equation of a line to solve for the y-intercept algebraically. Questions 14-15: The data in your experiment is governed by the formula: A = (BC/D) E You graph A on the y (vertical) axis and C on the x (horizontal) axis. 14. What is the slope theoretically equal to? a. B/D b. A/E c. E d. C/A e. C/D 15. What is the y-intercept theoretically equal to? a. B/D b. E c. E d. C/A e. C/D Uncertainty in Measurements Questions 16-17: You are using a ruler to measure lengths. Assume that you can reasonably measure to one-half of the smallest division. This means you can see if the length of the object is closer to one of the tick marks, half-way between two tick marks, or closer to the following tick mark. (If you can t tell if a measurement is closer to a tick mark or half-way in between, then you can only reasonably measure to the nearest whole of the smallest division). This is your instrumental precision (instrumental uncertainty). 16. If the smallest division on your ruler is 1 mm (= 0.1 cm), what is the numerical value of your instrumental precision (instrumental uncertainty)? a. 1 mm b. 0.1 cm c cm d mm 17. If a rod is exactly three centimeters long, how would you record the length to reflect the precision of the instrument (and therefore, how well you can measure the length)? a. 30 mm b. 3.0 cm c cm d cm Note: Most analog instruments can be reasonably read to the nearest ½ of the smallest spacing on the instrument (e.g., ½ mm on a standard metric ruler). If the tick marks are widely spaced, you may estimate to the nearest 1/3 of the smallest spacing. Measuring to ¼ or 1/10 of the smallest spacing would be unreasonable. Questions 18-19: A student made the following six measurements of the mass of an object: m = 56.6 g, 56.6 g, 56.6 g, 56.6 g, 56.6g, 56.6 g. The instrumental uncertainty is δ inst = 0.1 g. 18. What is a reasonable estimate of the sample uncertainty, δ samp? a. 0.0 g b. 0.1 g c. 0.6 g d g 19. What is a reasonable estimate of the uncertainty of the measurement, δm? a. 0.0 g b. 0.1 g c. 0.6 g d g
4 20. A student made the following nine measurements of the wavelength of a standing wave: l = 14.4 cm, 15.1 cm, 14.2 cm, 14.6 cm, 14.4 cm, 14.8 cm, 13.7 cm, 13.9 cm, 14.9 cm. Assume the instrumental uncertainty is 0.1 cm. The best estimate of the wavelength of the standing wave is approximately 14.4 cm. This means that the difference between the individual measurements and the best estimate are approximately (same order as above): 0.0 cm, +0.7 cm, -0.2 cm, +0.2 cm, 0.0 cm, +0.4 cm, -0.7 cm, -0.5 cm, and +0.5 cm. One reasonable estimate of the sample uncertainty is ± 0.7 cm (Covers all of the samples). Are there any other reasonable estimates of the sample uncertainty? (Choose the best answer.) a. Yes: ±0.1 cm is also reasonable. b. Yes: ±1 cm is also reasonable. c. Yes: ±0.5 cm is also reasonable. d. Yes: ±0.4 cm is also reasonable. e. No: ±0.7 cm is the only reasonable sample uncertainty. Questions 21-23: The following questions test your knowledge of some of the terminology used with the identification and propagation of uncertainty. Use the following x = (10.8 ± 0.4) cm 21. What is the best estimate of x? a cm b. 0.4 cm c. 0.4/10.8 = = 3.85% d to 11.2 cm 22. What is the absolute uncertainty of x? a cm b. 0.4 cm c. 0.4/10.8 = = 3.85% d to 11.2 cm 23. What is the relative uncertainty of x? a cm b. 0.4 cm c. 0.4/10.8 = = 3.85% d to 11.2 cm 24. A student measures a coefficient of linear expansion of (14.2 ± 0.8)x10-6 C -1 for a steel tube and the accepted value for commercially available steel ranges from 11.3 x10-6 C -1 to 13.5x10-6 C -1. How does the measured value compare to the accepted value? (Consider whether the ranges overlap. Is there any agreement at all, no matter how small?) a. They do not agree. b. They are close but do not quite agree. c. They agree within uncertainty. Questions 25-30: Propagating Uncertainty through a calculation Use the Rules for Propagating uncertainty (step-by-step method) to solve the following calculations including uncertainty. You will need to do similar calculations to some of those below in the analysis for several labs this quarter. Pay attention to units! It is usually better to convert units before you do the calculation. Remember to convert the uncertainty if you convert the best estimate. The final answer should be rounded properly following the Rules for reporting experimental values (Appendix C of the Phy 121L/131L lab manual OR see Do s and Don ts of Physics Lab Reports #26). Reduce rounding error by keeping at least two extra nonsignificant digits in intermediate calculations, and only round severely in the last step. A = (10.0 ± 0.1) mm; B = (5.00 ± 0.10) cm; C = (8.0 ± 0.1) mm; D = (4.2 ± 0.2) cm 25. Convert C to cm. a. (0.8 ± 0.1) cm b. (8.0 ± 1.0) cm c. (0.80 ± 0.01) cm
5 26. B + C =? a. (13.0 ± 0.2) cm b. (5.80 ±.11) cm c. (58.0 ± 0.2) mm 27. B D =? a. (0.8 ± 0.3) cm b. (0.80 ±.21) cm c. (0.8 ± 0.1) cm 28. ½D =? a. (4.2 ± 0.1) cm b. (2.10 ± 0.05) cm c. (2.04 ±.10) cm d. (2.1 ± 0.1) cm 29. AC =? a. (80.0 ± 0.02) mm 2 b. (80.0 ± 1.8) mm 2 c. (80.0 ±.2) mm B 2 =? a. (25.0 ± 1.0) cm 2 b. (10.0 ± 4.0) cm 2 c. (25.0 ± 0.2) cm 2 d. (25.0 ± 0.01) cm 2
6 Extra Credit: Propagating Uncertainty through a multistep calculation with hints *** Up to 10 points EC but a maximum score of 35 out of 30 on Review Assignment *** Do following multistep calculations including uncertainty. Write out your final answer (value with its associated uncertainty) following the Rules for reporting experimental values (see Appendix C of the Phy 121L/131L lab manual or Do s and Don ts #26). Reduce rounding error by keeping at least two extra non-significant digits in intermediate calculations. Show all work to receive credit, including converting absolute to relative uncertainties (%) or relative uncertainties to absolute uncertainties, where needed. 1EC. [(34.35 ± 0.05) g (30.35 ± 0.05) g] (9.80 m/s 2 ) =??? Answer = 2EC. [(7.73 ± 0.01) cm] [(19.0 ± 0.1)mm] 2 =??? Give final answer in cm 3. Hint: first step is to make both length measurements have the same units (mm cm). Answer =
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