Significant Figures2.4. Rewrite the following equations in their clearest and most appropriate forms: (a) x= 3.323 ± 1.4 mm (b) t =1,234,567 ± 54,321 s (c) A= 5.33 X 10^-7 ± 3.21 X 10^-9 m (d) r =0.000,000,538 ± 0.000,000,03 mm Comparison of Measured and Accepted Values 2.8. Two groups of students measure the charge of the electron and report their results as follows: Group A: e = (1.75 ± 0.04) X 1O^-19 C And Group B: e = (1.62 ± 0.04) X 10^-19 C What should each group report for the discrepancy between its value and the accepted value, e = 1.60 X 10^-19 C (with negligible uncertainty)? Draw an illustration similar to that in Figure 2.2 to show these results and the accepted value. Which of the results would you say is satisfactory? Object distances p (in cm) and corresponding focal lengths f (I cm).9.16. satisfying the lens equation. for Problem) 2. Make a plot of l against p.2.16. thin spherical lenses) can be characterized by a parameter called the focal length / and that if an object is placed at a distance p from the lens. 11/ = (lip) + (l/q). the lens forms an image at a distance q.9. You have learned (or will learn) in optics that certain lenses (namely. with appropriate error bars. Object distance p (negligible uncertainty) Focal length f 45 55 28 65 75 33 85 (all ± 2) 34 37 40 . She then calculates the corresponding values of l from the lens equation and obtains the results shown in Table 2. To check if these ideas apply to a certain lens. and decide if it is true that this particular lens has a unique focal length f Table 2. a student places a small light bulb at various distances p from the lens and measures the location q of the corresponding images. where / always has the same value for a given lens. 4 0. In particular. a student measures v2 and h for seven different throws and gets the results shown in Table 2. use squared paper. including vertical and horizontal error bars.6 m/s^2? Table 2.8 7 v^2 (m^2/s^2) 7±3 7±3 5±3 8±4 5±5 2±5 2±6 .8 1 1. The slopes of these lines give the largest and smallest probable values of the slope.05 0. label your axes. v2 should be proportional to h. To test this proportionality.4 6 3. draw what seems to be the best straight line through the points and then measure its slope. To find the slope.0 3 2.18 If a stone is thrown vertically upward with speed v. Are your results consistent with the accepted value 2g = 19. (a) Make a plot of v2 against h.4 2 2. it should rise to a height h given by v2 = 2gh.6 4 3. (As usual.11. and choose your scale sensibly.) Is your plot consistent with the prediction that v2 <X h? (b) The slope of your graph should be 2g.11 Heights and Speeds of a stone thrown Vertically upward h (m) all ± 0. draw the steepest and least steep lines that seem to fit the data reasonably. To find the uncertainty in the slope.2. (a) A student's calculator shows an answer 123.Significant Figures and Fractional Uncertainties 2. Note how. . find out which definition of the standard deviation your calculator uses. 12 (a) Find the mean and the standard deviation. (b) If you don't yet know how to calculate the mean and standard deviation using your calculator's built-in functions.123. adopt the convention that a number with N significant figures is uncertain by ± 1 in the nth digit. even with only three measurements. (c) Do the same for the number 321.26.9) (the sample standard deviation) and the original definition (4. in particular. Use your calculator to check your answers to part (a). the difference between the two definitions is not very big.6) (the population standard deviation). You measure the time for a ball to drop from a second-floor window three times and get the results (in tenths of a second): 11. For the latter. 13. take a few minutes to learn.321.1.) (b) Do the same for the number 1231. If the student decides that this number actually has only three significant figures. what are its absolute and fractional uncertainties? (To be definite. use both the "improved" definition (4. (d) Do the fractional uncertainties lie in the range expected for three significant figures? The Mean and Standard Deviation 4.23. 8 m/s2? 4.The Standard Deviation of the Mean 4. a student concludes that the standard deviation (Tu of her measurements is (Tu = 10 m/s. (a) How many measurements are needed to give a final uncertainty of ±3 m/s? (b) How many for a final uncertainty of only ±0.2. what should be the student's best estimate for g and its uncertainty? (b) How well does her result agree with the accepted value of 9.5 m/s? . she could get any desired precision by making enough measurements and averaging. If all uncertainties were truly random.18 After measuring the speed of sound u several times. (a) Based on the five measurements of g reported in Problem 4.16. 24. In some experiments. (d) Suggest a couple of ways he could modify the experiment to reduce the effect of this systematic error. and timer. neglect of heat losses from a badly insulated calorimeter or neglect of friction for a poorly lubricated cart. d (meters): 15. tape measure. you may often appropriately do just one for a representative case and assume that all five cases are reasonably similar. assuming that all errors are random? Show that this answer is inconsistent with the accepted value of g = 9.Systematic Errors 4.915 2. for example.] . (b) Based on these results.68 Time. and he suggests that air resistance is probably the culprit.804 1. He assumes that air resistance is negligible and that the distance fallen is given by d = ~g? Using a tape measure and an electric timer. (c) Having checked his calculations. what is his best estimate for g.43 17.80 m/s2. t (seconds): 1.043 2. he concludes (correctly) that there must be some systematic error causing an acceleration different from 9. instead of doing five separate error propagations.80 m/s2. he measures the distances and times of the four separate drops as follows: Distance. In other words.149 (a) Copy these data and add a third row in which you put the corresponding accelerations.62 21. Give at least two arguments to support this suggestion.37 19. it is appropriate (and time saving) in many experiments to assume they are at least approximately so. calculated as g = 2d/f. [Although the percent uncertainties in the five measurements of I are probably not exactly the same. Here is another example: A student wants to measure the acceleration of gravity g by timing the fall of a wooden ball (3 or 4 inches across) dropped from four different windows in a tall building. systematic errors can be caused by the neglect of an effect that is not (in the situation concerned) negligible. . (See Figure 4.669 1. He then calculates the mean of these five values.5%. T (s): 1.7 88.2 59. his measurements were indeed systematically off by the radius of the ball. however.) He records five different lengths of the pendulum l and the corresponding periods T as follows: Length. their SD.804 1.4.6 cm/s2 and is horrified to realize that his discrepancy is nearly 10 times larger than his uncertainty. Make the necessary corrections to his data and compute his final answer for g with its uncertainty.4. l (cm): 51. Here is an example: A student tries to measure g using a pendulum made of a steel ball suspended by a light string.3 Period. . He is sure there was no problem with the measurement of the period T. he realizes that 1 cm is about the radius of the ball and that the lengths he recorded were the lengths of the string.566 1. (d) This result would mean that his length measurements suffered a systematic error of about a centimeter-a conclusion he first rejects as absurd. so he asks himself the question: How large would a systematic error in the length l have to be so that the margins of the total error just included the accepted value 979. he takes the SDOM as his final uncertainty and quotes his answer in the standard form of mean ± SDOM. (c) Having checked all his calculations. he calculates g as g = 41t2l/T2. which turns out to be 2.28.7 68. and their SDOM. What is his answer for g? (b) He now compares his answer with the accepted value g = 979. As he stares at the pendulum. he concludes that he must have overlooked some systematic error.2 79. Because the correct length of the pendulum is the distance from the pivot to the center of the ball (see Figure 4. Assuming all his errors are random.448 1.4).00 cm.6 cm/s2? Show that the answer is approximately 1. Confirm this sad conclusion.896 (a) For each pair. He therefore uses callipers to find the ball's diameter. Systematic errors sometimes arise when the experimenter unwittingly measures the wrong quantity.