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Posted: Sun Jul 10, 2022 10:09 am
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3. The rate at which a sample of radioactive material emits radiation decreases exponentially as the material is depleted. This effect is observable on short time scales if the half-life of the substance is relatively short. To study this exponential decay and find the mean lifetime t, a student places a counter near a sample and records the number of decays in 15 seconds. She repeats the counting six times, at 30-minute intervals, and obtains the results in the following table. Notice that, because it takes about 10 minutes to prepare the equipment, her first measurement is made at t = 10 min. Table 2: Number v(t) of emissions in a 15-second period versus total time elapsed t (in min) t = elapsed time (min) 100 v = number of counts per 15s 60 If the sample does decay away exponentially, the counts v(t) should follow the relation v(t): = Voe-t/t 10 188 40 102 70 31.4 130 18 where t is the unknown mean lifetime of the material and vois another unknown constant representing the inferred count rate at t = 0. Excel is recommended for this analysis. 160 5
a) Plot the raw data to verify whether it roughly follows the expected exponential relationship. Does the data set look correct? If not, can you explain why and amend it? Hint: Most mistakes come from copying and pasting data in spreadsheets! (2 point) b) Decide whether considering uncertainty in t will be necessary, and add the uncertainties on the v; to Table 2. If they vary widely, especially if they follow a trend, we'll need to make a weighted least-squares fit. (2 point) c) Write down the linearization of the exponential model z(t) = ln (v(t)) and identify expressions for the slope and y-intercept. (1 point) d) Use the general rule for error propagation to show that the uncertainty on z(t) is 1/√v(t). Add the uncertainties on the z; to Table 2. (2 points) f) e) What are the weights w; of the linearized data z;? Add them to Table 2 also. (1 point) Obtain a best estimate of the mean lifetime t and its uncertainty using a weighted least- square fit. You may use software to sum lists and check your final results, but do not use a fitting algorithm to obtain your results. (3 points) Draw a graph using Excel and/or Python showing your (linearized) data with error bars and your weighted least-square best fit line. Looking at your graph, about how many decays would have occurred in the first 15 seconds after t = 0? Is this consistent with your value of vo? (3 points)
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a) Plot the raw data to verify whether it roughly follows the expected exponential relationship. Does the data set look correct? If not, can you explain why and amend it? Hint: Most mistakes come from copying and pasting data in spreadsheets! (2 point) b) Decide whether considering uncertainty in t will be necessary, and add the uncertainties on the v; to Table 2. If they vary widely, especially if they follow a trend, we'll need to make a weighted least-squares fit. (2 point) c) Write down the linearization of the exponential model z(t) = ln (v(t)) and identify expressions for the slope and y-intercept. (1 point) d) Use the general rule for error propagation to show that the uncertainty on z(t) is 1/√v(t). Add the uncertainties on the z; to Table 2. (2 points) f) e) What are the weights w; of the linearized data z;? Add them to Table 2 also. (1 point) Obtain a best estimate of the mean lifetime t and its uncertainty using a weighted least- square fit. You may use software to sum lists and check your final results, but do not use a fitting algorithm to obtain your results. (3 points) Draw a graph using Excel and/or Python showing your (linearized) data with error bars and your weighted least-square best fit line. Looking at your graph, about how many decays would have occurred in the first 15 seconds after t = 0? Is this consistent with your value of vo? (3 points)