# Experimental Uncertainties

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1 Experimental Uncertainties 1 Measurements of any physical quantity can never be exact. One can only know its value with a range of uncertainty. If an experimenter measures some quantity X, the measurement must be written with an uncertainty: X ± ΔX. This expresses the experimenter s judgment that the true value of X lies somewhere between X - ΔX and X + ΔX. I. Uncertainties of Measurements 1. Instrumental uncertainty Every measuring instrument has an inherent uncertainty that is determined by the precision of the instrument. This value is taken as a half of the smallest increment of the instrument scale. For example, if the smallest mark on a ruler is every 1 mm, then 0.5 millimeter is the precision of a ruler. What is the logic of this? Take a look at the two examples below: A measurement of 73 mm Where is the location of the end of the pencil on the ruler? If the smallest unit on the ruler is 1 mm, then we would say that the end is at 73 mm. If the end was further right than 73.5 mm, we would round up to 74 mm since that would be the closest mark on the ruler 70 mm 80 mm 73.5 mm A measurement of 74 mm In this case the end of the pencil is past the middle in between 73 mm and 74 mm. Thus when reading location of the end of the pencil on the ruler, we would round our measurement up to 74 mm since it is the closest mark on the ruler 70 mm 80 mm 73.5 mm

5 If the two values X and Y do not agree to within their experimental uncertainties, we say they are different. 2. Measured and Calculated Quantities Suppose you want to determine the uncertainty in the final value of a quantity that is calculated from several measured quantities. The uncertainties in these measured quantities propagate through the calculation to produce uncertainty in the final result. Consider the following example. Suppose you know the average mass of one apple m with the uncertainty Δm. If you want to calculate the mass M of the basket of 100 apples, you will get the value M ± ΔM = 100 m ± 100 Δm. The relative uncertainty of calculated value of M remains the same as the relative uncertainty of the single measured value for m ΔM / M = Δm / m. If you have more than one measured quantity, estimating uncertainty becomes a bit more complicated. The way we will handle it is with the weakest link rule. 3. Weakest link rule The percent uncertainty in the calculated value of some quantity is at least as great as the greatest percentage uncertainty of the values used to make calculation. Thus to estimate uncertainty in you calculated value, you have to: 1. Estimate the absolute uncertainty in each measured quantity used to find the calculated quantity. 2. Calculate the relative uncertainty in each measured quantity. 3. Pick the largest relative uncertainty. We call this largest relative uncertainty the weakest link. 4. We say that the relative uncertainty in our calculated value is equal to the weakest link (the largest relative uncertainty in our measured values). We can then apply the relative uncertainty of the weakest link to the calculated quantity to determine its absolute uncertainty. Here s an example: You ve been asked to estimate the volume of your laptop computer. First, you measure the length, width, and thickness with a meter stick (which has an absolute uncertainty of 0.05cm) Measurement Value (with absolute uncertainty) Relative uncertainty Length (39.4 ± 0.05) cm 0.05 cm 39.4 cm =1.27 "10#3 = 0.127% Width Thickness X-!X [ X X-!X [ X+!X ] X (28.7 ± 0.05) cm (4.3 ± 0.05) cm X+!X [ ] ] Y-!Y 0.05 cm 28.7 cm =1.74 "10#3 = 0.174% 0.05 cm 4.3 cm =11.6 "10#3 =1.16% Y Y-!Y Y+!Y Y [ ] Y+!Y 5

6 6 From this table, you can see that the thickness has by far the largest relative uncertainty - the thickness measurement is our weakest link! The volume of the laptop is V = LWT = (39.4 cm)(28.7 cm)(4.3 cm) = 4862 cm 3 Since the thickness measurement has the largest relative uncertainty (1.16%) we say this is the relative uncertainty in our final calculated volume V. To determine the absolute uncertainty of our calculated volume, we multiply the volume by the relative uncertainty of the weakest link:!v = (4862 cm 3 )(1.16 "10 #2 ) = 56 cm 3 So, the final estimate for the volume of the laptop is V = (4862 ± 56) cm Comparable uncertainties If a final calculated value depends on several measured quantities that each has comparable relative uncertainties, then the rules are more complicated: they depend on the type of the mathematical relationship you use for the calculation. Keep in mind, however, that the overall relative uncertainty cannot be less than the relative uncertainties of the independent measured quantities. In other words, no matter what the relationship is, the relative uncertainty can only increase when doing calculations. Consider a simple example first: Let a calculated value C be the produce of two measured quantities, A and B. In other words: C=AB. Now let s rewrite A and B to include their " uncertainties: A ±!A = A 1±!A % \$ ' = A(1±!A). Likewise, B ±!B = B(1±!B). Now we can # A & estimate the product of A and B as AB(1±!A)(1±!B) = AB(1±!A ±!B ±!A!B). If we assume that the relative uncertainties δa and δb are much smaller than 1, then we can neglect the δaδb term and the following statement is approximately true: C(1±!C) = AB(1±!A ±!B) Thus the relative uncertainty δc in the measured quantity C is approximately:!c =!A +!B. The relative uncertainty of a product of measured values is approximately equal to the sum of the individual relative uncertainties if those relative uncertainties are roughly equal. A useful consequence of this is that if C=A 2 =A.A, then:!c =!A +!A = 2!A For instance, if you measure the sides of a square with an uncertainty of 3%, the resulting uncertainty in its area would be 6%. On a side-note, we can observe something very interesting if we do the same analysis in terms of absolute uncertainties instead of relative uncertainties:!c " C #!C = AB(!A +!B) " B #!A + A #!B In other words,!(ab) = B "!A + A "!B, which very much resembles the product rule for derivatives! Here is a trickier example. Suppose you measured the period of a pendulum with an uncertainty 3% and want to calculate the pendulum s frequency. What would be its percent uncertainty? First recall that f=1/t. Now as the period is T(1±δT), the estimate for the frequency will be

7 1 1 (1 δt ) = f (1 δt ) T(1 ± δt ) T, as you might remember from calculus. Although the sign of the percent uncertainty is reversed on the righthand side, it is practically the same relative uncertainty! (δf = δt) Thus the frequency will also be determined with an uncertainty of 3%. In general, the percent uncertainty of a measured quantity equals the percent uncertainty of its calculated inverse. 7 IV Summary When you are doing a lab and measuring some quantities to determine an unknown quantity: Decide which factors affect your result the most. Wherever possible, try to reduce the effects of these factors that cause uncertainty. Wherever possible, try to reduce uncertainties by measuring longer distances or time intervals, etc. Decide the absolute uncertainty of each measurement. Then, find the relative uncertainty of each measurement. If one relative uncertainty is much larger than the others, you can ignore all other sources and use this uncertainty to write the value of the relative uncertainty of the quantity that you are calculating. If relative uncertainties of several measured quantities are comparable, the overall relative uncertainty can be computed as follows: o Multiplication by a constant does not affect the relative uncertainty o When two or more measured quantities are multiplied (or divided), the relative uncertainties should be added up. In particular, when raising a measured quantity to a power (e.g., n=3), the percent uncertainty should be multiplied by the corresponding factor (e.g., 3). o When two or more measured quantities are added/subtracted, their absolute uncertainties add up. Find the range where your calculated quantity lies. Make a judgment about your results taking into the account this uncertainty.

9 Solutions to Exercises 9 1. Relative uncertainties δf and δt are both dimensionless quantities since they are expressed as ratios:! f =!f and!t =!T. So yes, units are indeed consistent since there aren t any. f T 2.!D =!A +!B +!C. 3.!C =!A +!B. 4. V = 4 3! R3. Therefore δv=3δr=6%. 5. The percent uncertainty, also called relative uncertainty, is 0.25 m = =10.6% m 6 miles 6. Your average speed is = 2.4 miles per hour. The relative uncertainty in the distance 2.5 hours 0.1 miles measurement is = =1.67%. The absolute uncertainty in the time measurement 6miles is 0.5 hours since the hike might have taken as short as 2 hours or as long as 3 hours. The relative 0.5 hours uncertainty in the time measurement is = 0.2 = 20%. The distance measurement is 2.5 hours more accurate because its relative uncertainty is smaller. Because it is significantly smaller the weakest link may be used to estimate the uncertainty in the speed. The weakest link rule says to use the largest relative uncertainty from the data as the relative uncertainty of anything you calculate from the data. So, the relative uncertainty in the speed is 20%. The absolute uncertainty in the speed is then 2.4 miles per hour " 0.2 = 0.48 miles per hour. 0.5 s s s 7. The average time of the fall is 0.5 s. We can estimate the absolute 3 uncertainty by looking at the range of measured values. The absolute uncertainty is 0.1 s since all the trials lie within 0.1s of the average. The relative uncertainty is 0.1 s = 0.2 = 20%. 0.5 s 8. The average is 1.6 s and the relative uncertainty is 0.1 s = = 6.25%. Conclusion: The 1.6 s measurement of the time of fall from 10 m is more accurate than the time of fall from 1m since the relative uncertainty is lower. 9. This is a little trickier. Each distance measurement could reasonably have an absolute uncertainty of 0.5 miles. That is: (210 ± 0.5) miles and (260 ± 0.5) miles. The distance measured is 260 miles miles = 50 miles. However, because of the uncertainty in each measurement, the interval could be as large as miles " miles = 51 miles, and could be as small as miles " miles = 49 miles. Thus the distance interval measured is (50 ±1) miles. The absolute uncertainty is 1 mile. The relative uncertainty is 1 mile = 0.02 = 2%. A reasonable estimate of the absolute uncertainty in the time 50 miles

10 0.5 min measurement is 0.5 min. Its relative uncertainty is = =1.11%. The average speed 45 min is the distance traveled (50 miles) divided by the time it took (45 min = 0.75 hours) 50 miles v avg = = 66.7 miles/hour. What is the uncertainty in our estimate of the average 0.75 hours speed? Because the two relative uncertainties in the distance and time measurements are comparable to each other we should add them to find the relative uncertainty in the calculated average speed. The relative uncertainty in the speed is 1.11%+2%=3.11%. So the absolute uncertainty is 66.7 miles/hour x = 2.1 miles/hour. The average speed is then (66.7 ± 2.1) miles/hour. 10. The temperature drop is 76 C - 68 C = 8.0 C. The process we need to use here is similar to problem 5. Remember, our measurement is the drop: 8.0 C, but to find the uncertainty in this measurement we need to know the uncertainty in each temperature, 76 C and 68 C. We know that the temperature measurements are (76 ±1) C and (68 ±1) C. Thus our measurement of the temperature drop could have been as large as 77 C " 67 C =10 C and as small as 75 C " 69 C = 6 C. This makes the absolute uncertainty of the temperature drop is 2 C. Based on the above analysis the temperature drop is (8 ± 2) C. 10

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