Significant Figures 2.4. 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?
2.16. You have learned (or will learn) in optics that certain lenses (namely, 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, the lens forms an image at a distance q, satisfying the lens equation, 11/ = (lip) + (l/q), where / always has the same value for a given lens. To check if these ideas apply to a certain lens, a student places a small light bulb at various distances p from the lens and measures the location q of the corresponding images. She then calculates the corresponding values of l from the lens equation and obtains the results shown in Table 2.9. Make a plot of l against p, with appropriate error bars, and decide if it is true that this particular lens has a unique focal length f Table 2.9. Object distances p (in cm) and corresponding focal lengths
f (I cm); for
Problem) 2.16. Object distance p (negligible uncertainty)
Focal length f
45 55
28
65 75
33
85
(all ± 2) 34 37 40
2.18 If a stone is thrown vertically upward with speed v, it should rise to a height h given by v2 = 2gh. In particular, v2 should be proportional to h. To test this proportionality, a student measures v2 and h for seven different throws and gets the results shown in Table 2.11. (a) Make a plot of v2 against h, including vertical and horizontal error bars. (As usual, use squared paper, label your axes, and choose your scale sensibly.) Is your plot consistent with the prediction that v2
Heights and Speeds of a stone thrown Vertically upward
h (m) all ± 0.05 0.4 0.8 1 1.4 2 2.0 3 2.6 4 3.4 6 3.8 7
v^2 (m^2/s^2) 7±3 7±3 5±3 8±4 5±5 2±5 2±6
Significant Figures and Fractional Uncertainties 2.26. (a) A student's calculator shows an answer 123.123. If the student decides that this number actually has only three significant figures, what are its absolute and fractional uncertainties? (To be definite, adopt the convention that a number with N significant figures is uncertain by ± 1 in the nth digit.) (b) Do the same for the number 1231.23. (c) Do the same for the number 321.321. (d) Do the fractional uncertainties lie in the range expected for three significant figures?
The Mean and Standard Deviation 4.1. 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, 13, 12 (a) Find the mean and the standard deviation. For the latter, use both the "improved" definition (4.9) (the sample standard deviation) and the original definition (4.6) (the population standard deviation). Note how, even with only three measurements, the difference between the two definitions is not very big. (b) If you don't yet know how to calculate the mean and standard deviation using your calculator's built-in functions, take a few minutes to learn. Use your calculator to check your answers to part (a); in particular, find out which definition of the standard deviation your calculator uses.
The Standard Deviation of the Mean 4.16. (a) Based on the five measurements of g reported in Problem 4.2, 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 9.8 m/s2?
4.18 After measuring the speed of sound u several times, a student concludes that the standard deviation (Tu of her measurements is (Tu = 10 m/s. If all uncertainties were truly random, she could get any desired precision by making enough measurements and averaging. (a) How many measurements are needed to give a final uncertainty of ±3 m/s? (b) How many for a final uncertainty of only ±0.5 m/s?
Systematic Errors
4.24. In some experiments, systematic errors can be caused by the neglect of an effect that is not (in the situation concerned) negligible, for example, neglect of heat losses from a badly insulated calorimeter or neglect of friction for a poorly lubricated cart. 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. 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, he measures the distances and times of the four separate drops as follows: Distance, d (meters): 15.43 17.37 19.62 21.68 Time, t (seconds): 1.804 1.915 2.043 2.149
(a) Copy these data and add a third row in which you put the corresponding accelerations, calculated as g = 2d/f. (b) Based on these results, what is his best estimate for g, assuming that all errors are random? Show that this answer is inconsistent with the accepted value of g = 9.80 m/s2. (c) Having checked his calculations, tape measure, and timer, he concludes (correctly) that there must be some systematic error causing an acceleration different from 9.80 m/s2, and he suggests that air resistance is probably the culprit. Give at least two arguments to support this suggestion. (d) Suggest a couple of ways he could modify the experiment to reduce the effect of this systematic error. [Although the percent uncertainties in the five measurements of I are probably not exactly the same, it is appropriate (and time saving) in many experiments to assume they are at least approximately so. In other words, instead of doing five separate error propagations, you may often appropriately do just one for a representative case and assume that all five cases are reasonably similar.]
4.28. Systematic errors sometimes arise when the experimenter unwittingly measures the wrong quantity. Here is an example: A student tries to measure g using a pendulum made of a steel ball suspended by a light string. (See Figure 4.4.) He records five different lengths of the pendulum l and the corresponding periods T as follows:
Length, l (cm): 51.2 59.7 68.2 79.7 88.3 Period, T (s): 1.448 1.566 1.669 1.804 1.896 (a) For each pair, he calculates g as g = 41t2l/T2. He then calculates the mean of these five values, their SD, and their SDOM. Assuming all his errors are random, he takes the SDOM as his final uncertainty and quotes his answer in the standard form of mean ± SDOM. What is his answer for g? (b) He now compares his answer with the accepted value g = 979.6 cm/s2 and is horrified to realize that his discrepancy is nearly 10 times larger than his uncertainty. Confirm this sad conclusion. (c) Having checked all his calculations, he concludes that he must have overlooked some systematic error. He is sure there was no problem with the measurement of the period T, 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 979.6 cm/s2? Show that the answer is approximately 1.5%. (d) This result would mean that his length measurements suffered a systematic error of about a centimeter-a conclusion he first rejects as absurd. As he stares at the pendulum, however, he realizes that 1 cm is about the radius of the ball and that the lengths he recorded were the lengths of the string. Because the correct length of the pendulum is the distance from the pivot to the center of the ball (see Figure 4.4), his measurements were indeed systematically off by the radius of the ball. He therefore uses callipers to find the ball's diameter, which turns out to be 2.00 cm. Make the necessary corrections to his data and compute his final answer for g with its uncertainty.
