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MSU PHY 252 - appendixA

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APPENDIX A: DEALING WITH UNCERTAINTY 1. OVERVIEW • An uncertainty is always a positive number δx > 0. • If the uncertainty of x is 5%, then δx = .05x. • If the uncertainty in x is δx, then the fractional uncertainty in x is δx/x. • If you measure x with a device that has a precision of u, then δx is at least as large as u. • The uncertainty of x + y or x - y is δx + δy. • If d is data and e is expectation: The difference is Δ = d - e % difference is 100 (Δ / e) They are compatible IF |d - e | < δd + δe • Fractional uncertainty of z = xy or x / y is: δz / z = δx / x + δy / y • Fractional uncertainty of f = xp yq zr is: δf / f = pδx / x + qδy / y + rδz/z • Uncertainty of f(x) is: |f(x+δx) – 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): L0 = 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) Manufacturer’s tolerance Suppose I purchase a nominally 100 Ω resistor from a manufacturer. It has a gold band on it which signifies a 5% tolerance. What does this mean? The tolerance means δR/R = 0.05 = 5%, that is, the fractional uncertainty. Thus, δR = R x 0.05 = 5Ω. We write this as R = Rnominal ± δR = 100 ± 5 Ω. It says that the company certifies that the true resistance R lies between 95 and 105Ω. That is, 95 ≤ R≤ 105Ω. 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. (c) Reading a digital meter. Suppose I measure the voltage across a resistor using a digital multimeter. The display says 7.45 V and doesn’t change as I watch it. The general rule is that the uncertainty is half of the value of the least significant digit. This value is 0.01 V so that half of it is 0.005. Here’s why. The meter can only display two digits to the right of the decimal so it must round off additional digits. So if the true value is between 7.445 V and 7.454 V, the display will get rounded to 7.45 V. Thus the average value and its uncertainty can be written as 7.45 ± 0.005V. When you record this in your notebook, be sure to write 7.45 V. Not 7.450 V. Writing 7.450 V means that the uncertainty is 0.0005 V. Note that in this example we assumed that the meter reading is steady. If instead, the meter reading is fluctuating, then the situation is different. Now, you need to estimate the range over which the display is fluctuating, then estimate the average value. If the display is fluctuating between 5.4 and 5.8 V, you would record your reading as 5.6 ± 0.2 V. The uncertainty due to the noisy reading is much larger than your ability to read the last digit on the display, so you record the larger error. 3. USING UNCERTAINTIES IN CALCULATIONS We need to combine uncertainties so that the error bars almost certainly include the true value. (a) Adding and Subtracting Let’s look at the most basic case. We measure x and y and want to find the error in z. If z = x + y δz = δx + δy If z = x - yδz = δx + δy ♦ Note that the uncertainty for subtracting has exactly the same form as for adding. ♦ The most important errors are simply the biggest ones. Example: (7 ± 1 kg) – (5 ± 1 kg) = 2 ± 2 kg (b) Multiplying and Dividing If: a = b x c, the rule is: δaa = δbb + δcc If you need δa, use δa = a δbb+δcc⎛ ⎝ ⎞ ⎠ . For dividing, w =xy , the rule is the same as for multiplication: δ ww = δ xx + δ yy For δw, use δw = wδ xx+δ yy⎛ ⎝ ⎜ ⎞ ⎠ ⎟ . ♦ It is simplest to just remember the single boxed rule for multiplication and division. ♦ If the expression contains a constant, c, it has δc=0. ♦ The most important errors in multiplication and division are the largest fractional errors, not absolute errors. This makes sense if you consider that b and c need not have the same units - there is no way to compare the absolute sizes of quantities with different units. Example: V = IR I = 7 ± 1 mA R = 20 ± 2 Ω V = 140 mAΩ = 140 mV = 0.14 V δ VV = δ II + δ RR = 1 mA7 mA + 2 Ω20 Ω = 0.24 δV = 0.24 x 0.14 V = 0.034 V = 34 mVOur formula for multiplication indicates that multiplying by a perfectly known constant has no effect on the fractional error of a quantity: F = mg m = 12 ± 1kg g = 9.8 m/s F = 117.6 N δ FF = δ mm + δ gg = δ mm since δg = 0 Then δ FF = 1 kg12 kg and F = 117.6 ± 9.8 N The uncertainty δF = Fδ mm⎛ ⎝ ⎞ ⎠ = mgδ mm⎛ ⎝ ⎞ ⎠ = gδm. So, δF is just the constant g times δm. (c) Multiples If f = cx + dy + gz where c, d, and g are positive or negative constants then, δ(cx) = |cδx| δ(dy) = |dδy| δ(gz) = |gδz| from the multiplication rule. From the addition rule, δf = |cδx| + |dδy| + |gδz|. (d) Powers If f = xp yq zr where p, q, and r are positive or negative constants. ()()()zz r + yy q +x xp =ff rzrz + y y x x = ff qqppdδδδδδδδ+ (e) General Suppose we want to calculate f(x), a function of x, which has uncertainty δx. What is the uncertainty in the calculated value f? We simply calculate f at x, and again at x′ = x + δx, then take the absolute value of the difference: δf …


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MSU PHY 252 - appendixA

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