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Unformatted text preview: APPENDIX A: DEALING WITH UNCERTAINTY 1. OVERVIEW An uncertainty is always a positive number x > 0 . If the uncertainty of x is 5%, then x = .05x. If the uncertainty in x is x , then the fractional uncertainty in x is x/x. If you measure x with a device that has a precision of u , then x is at least as large as u . The uncertainty of x + y or x - y is x + y . If d is data and e is expectation: The difference is = d - e % difference is 100 ( / e) They are compatible IF |d - e | < d + e Fractional uncertainty of z = xy or x / y is: z / z = x / x + y / y Fractional uncertainty of f = x p y q z r is: f / f = p x / x + q y / y + r z/z Uncertainty of f(x) is: |f(x+ x) f(x)| 2. ESTIMATING UNCERTAINTIES FOR MEASURED QUANTITIES (a) Simple Measurements: The smallest division estimate Suppose we use a meter stick ruled in centimeters and millimeters, and you are asked to measure the length of a rod and obtain the results (see figure 1a) : L = 5.73 cm. A good estimate of the uncertainty here is half of the smallest division on the scale, or 0.05 cm. That is, the length of the rod would be specified as: L = 5.73 0.05cm This says that you are very confident that the length of the rod falls in the range 5.73 cm 0.05 cm to 5.73 cm + 0.05 cm, or the length falls in the range of 5.68 cm to 5.78 cm (see figure 1b). (b) Manufacturers tolerance Suppose I purchase a nominally 100 resistor from a manufacturer. It has a gold band on it which signifies a 5% tolerance. What does this mean? The tolerance means R/R = 0.05 = 5%, that is, the fractional uncertainty. Thus, R = R x 0.05 = 5 . We write this as R = R nominal R = 100 5 . It says that the company certifies that the true resistance R lies between 95 and 105 . That is, 95 R 105 . The company tests all of its resistors and if they fall outside of the tolerance limits the resistors are discarded. If your resistor is measured to be outside of the limits, either (a) the manufacturer made a mistake (b) you made a mistake or (c) the manufacturer shipped the correct value but something happened to the resistor that caused its value to change....
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- Summer '08