DATA ANALYSIS AND ERROR ESTIMATION Purpose To learn how to analyze experimental data and to practice error analysis. Error Estimation Suppose that a quantity X is measured with an uncertainty in the measurement ∆X. Then the measurement is recorded as X ± ∆X. ∆X is called the maximum possible error (MPE) or uncertainty inX. (∆X/X) is called the fractional error (or relative error) in X and (∆X/X)×100 % is the percent error. Now, suppose a quantity Z is a function of two independent quantities (variables) X and Y. The expressions for ∆Z for some common relations between X, Y and Z are given in the following table: Relation between X, Y and Z Relation between errors Z = X + Y or Z = X − Y ∆Z = ∆X + ∆Y Z = XY or Z = X/Y Z = Constant. XmYn (∆Ζ/Ζ)=(∆X/X) + (∆Y/Y) (∆Ζ/Ζ)= m(∆X/X) + n(∆Y/Y) or Z = Constant. Xm/Yn Exercise (1) Two lengths X and Y are measured with a meter ruler as follows: X = 35.5 ± 0.1 cm and Y = 67.3 ± 0.1 cm. Find the percent error in a) Z1 = X + Y b) Z2 = Y – X c) Z3 = XY d) Z4 = X/Y e) Z5 = X2/Y3 Exercise (2) The data in the following table relates to measurements made of the period of oscillation T of a simple pendulum of variable length L. Theory predicts the following relationship: where g is the acceleration due to gravity. = 2� 1. Calculate the value of g for each data point using the above equation. Observe the rules of significant figures when doing the calculation. Table 1 Length (L) cm ± 0.5 cm Period (T) sec ± 0.01 sec 57.5 50.0 41.5 35.0 29.0 22.5 16.5 1.51 1.42 1.28 1.19 1.06 0.93 0.81 g (cm/s2) T2 (sec2) 2. Write down the general expression for the fractional error (∆g/g) in terms of (∆T/T) and (∆L/L). 3. Calculate ∆g for the last data point in Table 1. 4. Record the value of g from the last data point with its error as g1 ± ∆g1 5. Find the average value of g for all data points in Table 1. Call this value gav. 6. Look at the g values in Table 1 and identify the maximum and minimum values of g (gmax and gmin). Make an estimate for the maximum possible error in g (call it ∆gav) from the following formula: ∆ = − 2 7. Record your experimentally determined value of g asgav ± ∆gav 8. Calculate T2 for each data point in Table 1, and plot T2 versus L. Determine g from the slope of the graph. 9. The accepted value of g in Dhahran area is 980 ± 1 cm/s2. Fill the table below and observe how the three different values of g agree (or disagree) with the accepted value and with each other within their uncertainty limits. Table 2 Accepted value From Step 4 From Step 7 From Step 8 g (cm/s2) ∆g (cm/s2) % error (∆g/g)×100 Lower bound (g−∆g) (cm/s2) Upper bound (g+∆g) (cm/s2) 980 1 0.1 979 981

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